# Recursive Derivative Orders, Exact Polynomial Coefficients and Existence Boundaries

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261111-r02-higher-order-derivative-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[1, 2, 3, 4] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(3, 0, Yes, f_zero_equals_f, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 0, Yes, f_zero_equals_f, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0001/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 2. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[2, 4, 6, 8, 10] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(4, 1, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 1, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0002/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 3. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[3, 6, 9, 12, 15, 18] for f(t)=sum coefficients[k]*t^k and derivative order=2≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(5, 2, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 2, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0003/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 4. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[4, 8, 12, 16, 20, 24, 28] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(6, 3, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 3, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0004/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 5. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[5, 10, 15, 20] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(3, 4, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 4, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0005/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 6. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[6, 12, 18, 24, 30] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(4, 5, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 5, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0006/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 7. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[7, 14, 21, 28, 35, 42] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(5, 6, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 6, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0007/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 8. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[8, 16, 24, 32, 40, 48, 56] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(6, 7, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 7, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0008/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 9. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[9, 18, 27, 36] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(3, 3, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 3, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0009/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 10. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[10, 20, 30, 40, 50] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(4, 3, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 3, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0010/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 11. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[11, 22, 33, 44, 55, 66] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(5, 3, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 3, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0011/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 12. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[12, 24, 36, 48, 60, 72, 84] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(6, 3, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 3, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0012/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 13. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[13, 26, 39, 52] for f(t)=sum coefficients[k]*t^k and derivative order=2≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(3, 2, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 2, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0013/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 14. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[14, 28, 42, 56, 70] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(4, 1, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 1, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0014/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 15. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[15, 30, 45, 60, 75, 90] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(5, 0, Yes, f_zero_equals_f, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 0, Yes, f_zero_equals_f, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0015/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 16. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[16, 32, 48, 64, 80, 96, 112] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(6, 7, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 7, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0016/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 17. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[17, 34, 51, 68] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(3, 1, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 1, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0017/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 18. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[18, 36, 54, 72, 90] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(4, 5, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 5, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0018/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 19. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[19, 38, 57, 76, 95, 114] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(5, 4, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 4, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0019/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 20. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[20, 40, 60, 80, 100, 120, 140] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(6, 3, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 3, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0020/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 21. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[21, 42, 63, 84] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(3, 0, Yes, f_zero_equals_f, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 0, Yes, f_zero_equals_f, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0021/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 22. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[22, 44, 66, 88, 110] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(4, 3, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 3, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0022/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 23. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[23, 46, 69, 92, 115, 138] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(5, 1, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 1, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0023/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 24. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[24, 48, 72, 96, 120, 144, 168] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(6, 7, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 7, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0024/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 25. Order Zero Is the Original Function; Later Orders Need the Recursive Chain

Use original low-to-high coefficients=[25, 50, 75, 100] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).

declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.

View answer and explanation(3, 4, Yes, recursive_derivative_chain_required, Yes)

Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 4, Yes, recursive_derivative_chain_required, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0025/)

Higher-order derivative · qf-mod42-higher-order-qualification · candidate





## 26. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[1, 2, 3, 4] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([1, 2, 3, 4], 0, [1, 2, 3, 4], -2, -23, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([1, 2, 3, 4], 0, [1, 2, 3, 4], -2, -23, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0026/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 27. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[2, 4, 6, 8, 10] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([2, 4, 6, 8, 10], 1, [4, 12, 24, 40], -1, -24, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([2, 4, 6, 8, 10], 1, [4, 12, 24, 40], -1, -24, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0027/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 28. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[3, 6, 9, 12, 15, 18] for f(t)=sum coefficients[k]*t^k and derivative order=2≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([3, 6, 9, 12, 15, 18], 2, [18, 72, 180, 360], 0, 18, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([3, 6, 9, 12, 15, 18], 2, [18, 72, 180, 360], 0, 18, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0028/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 29. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[4, 8, 12, 16, 20, 24, 28] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([4, 8, 12, 16, 20, 24, 28], 3, [96, 480, 1440, 3360], 1, 5376, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([4, 8, 12, 16, 20, 24, 28], 3, [96, 480, 1440, 3360], 1, 5376, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0029/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 30. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[5, 10, 15, 20] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([5, 10, 15, 20], 4, [0], 2, 0, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([5, 10, 15, 20], 4, [0], 2, 0, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0030/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 31. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[6, 12, 18, 24, 30] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([6, 12, 18, 24, 30], 5, [0], -2, 0, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([6, 12, 18, 24, 30], 5, [0], -2, 0, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0031/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 32. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[7, 14, 21, 28, 35, 42] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([7, 14, 21, 28, 35, 42], 6, [0], -1, 0, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([7, 14, 21, 28, 35, 42], 6, [0], -1, 0, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0032/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 33. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[8, 16, 24, 32, 40, 48, 56] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([8, 16, 24, 32, 40, 48, 56], 7, [0], 0, 0, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([8, 16, 24, 32, 40, 48, 56], 7, [0], 0, 0, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0033/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 34. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[9, 18, 27, 36] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([9, 18, 27, 36], 3, [216], 1, 216, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([9, 18, 27, 36], 3, [216], 1, 216, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0034/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 35. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[10, 20, 30, 40, 50] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([10, 20, 30, 40, 50], 3, [240, 1200], 2, 2640, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([10, 20, 30, 40, 50], 3, [240, 1200], 2, 2640, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0035/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 36. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[11, 22, 33, 44, 55, 66] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([11, 22, 33, 44, 55, 66], 3, [264, 1320, 3960], -2, 13464, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([11, 22, 33, 44, 55, 66], 3, [264, 1320, 3960], -2, 13464, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0036/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 37. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[12, 24, 36, 48, 60, 72, 84] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([12, 24, 36, 48, 60, 72, 84], 3, [288, 1440, 4320, 10080], -1, -6912, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([12, 24, 36, 48, 60, 72, 84], 3, [288, 1440, 4320, 10080], -1, -6912, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0037/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 38. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[13, 26, 39, 52] for f(t)=sum coefficients[k]*t^k and derivative order=2≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([13, 26, 39, 52], 2, [78, 312], 0, 78, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([13, 26, 39, 52], 2, [78, 312], 0, 78, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0038/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 39. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[14, 28, 42, 56, 70] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([14, 28, 42, 56, 70], 1, [28, 84, 168, 280], 1, 560, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([14, 28, 42, 56, 70], 1, [28, 84, 168, 280], 1, 560, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0039/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 40. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[15, 30, 45, 60, 75, 90] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([15, 30, 45, 60, 75, 90], 0, [15, 30, 45, 60, 75, 90], 2, 4815, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([15, 30, 45, 60, 75, 90], 0, [15, 30, 45, 60, 75, 90], 2, 4815, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0040/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 41. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[16, 32, 48, 64, 80, 96, 112] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([16, 32, 48, 64, 80, 96, 112], 7, [0], -2, 0, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([16, 32, 48, 64, 80, 96, 112], 7, [0], -2, 0, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0041/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 42. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[17, 34, 51, 68] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([17, 34, 51, 68], 1, [34, 102, 204], -1, 136, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([17, 34, 51, 68], 1, [34, 102, 204], -1, 136, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0042/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 43. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[18, 36, 54, 72, 90] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([18, 36, 54, 72, 90], 5, [0], 0, 0, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([18, 36, 54, 72, 90], 5, [0], 0, 0, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0043/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 44. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[19, 38, 57, 76, 95, 114] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([19, 38, 57, 76, 95, 114], 4, [2280, 13680], 1, 15960, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([19, 38, 57, 76, 95, 114], 4, [2280, 13680], 1, 15960, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0044/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 45. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[20, 40, 60, 80, 100, 120, 140] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([20, 40, 60, 80, 100, 120, 140], 3, [480, 2400, 7200, 16800], 2, 168480, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([20, 40, 60, 80, 100, 120, 140], 3, [480, 2400, 7200, 16800], 2, 168480, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0045/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 46. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[21, 42, 63, 84] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([21, 42, 63, 84], 0, [21, 42, 63, 84], -2, -483, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([21, 42, 63, 84], 0, [21, 42, 63, 84], -2, -483, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0046/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 47. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[22, 44, 66, 88, 110] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([22, 44, 66, 88, 110], 3, [528, 2640], -1, -2112, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([22, 44, 66, 88, 110], 3, [528, 2640], -1, -2112, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0047/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 48. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[23, 46, 69, 92, 115, 138] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([23, 46, 69, 92, 115, 138], 1, [46, 138, 276, 460, 690], 0, 46, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([23, 46, 69, 92, 115, 138], 1, [46, 138, 276, 460, 690], 0, 46, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0048/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 49. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[24, 48, 72, 96, 120, 144, 168] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([24, 48, 72, 96, 120, 144, 168], 7, [0], 1, 0, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([24, 48, 72, 96, 120, 144, 168], 7, [0], 1, 0, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0049/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 50. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Use original low-to-high coefficients=[25, 50, 75, 100] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (coefficients, order, derivative_coefficients, point, value, certificate).

View answer and explanation([25, 50, 75, 100], 4, [0], 2, 0, falling_factorial_coefficients_exact)

Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([25, 50, 75, 100], 4, [0], 2, 0, falling_factorial_coefficients_exact).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0050/)

Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate





## 51. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[1, 2, 3, 4] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(3, 4, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 4, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0051/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 52. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[2, 4, 6, 8, 10] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(4, 6, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 6, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0052/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 53. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[3, 6, 9, 12, 15, 18] for f(t)=sum coefficients[k]*t^k and derivative order=8≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(5, 8, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 8, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0053/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 54. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[4, 8, 12, 16, 20, 24, 28] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(6, 7, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 7, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0054/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 55. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[5, 10, 15, 20] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(3, 5, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 5, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0055/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 56. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[6, 12, 18, 24, 30] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(4, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0056/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 57. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[7, 14, 21, 28, 35, 42] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(5, 6, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 6, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0057/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 58. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[8, 16, 24, 32, 40, 48, 56] for f(t)=sum coefficients[k]*t^k and derivative order=8≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(6, 8, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 8, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0058/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 59. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[9, 18, 27, 36] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(3, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0059/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 60. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[10, 20, 30, 40, 50] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(4, 5, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 5, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0060/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 61. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[11, 22, 33, 44, 55, 66] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(5, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0061/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 62. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[12, 24, 36, 48, 60, 72, 84] for f(t)=sum coefficients[k]*t^k and derivative order=9≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(6, 9, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 9, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0062/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 63. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[13, 26, 39, 52] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(3, 4, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 4, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0063/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 64. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[14, 28, 42, 56, 70] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(4, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0064/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 65. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[15, 30, 45, 60, 75, 90] for f(t)=sum coefficients[k]*t^k and derivative order=8≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(5, 8, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 8, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0065/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 66. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[16, 32, 48, 64, 80, 96, 112] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(6, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0066/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 67. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[17, 34, 51, 68] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(3, 5, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 5, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0067/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 68. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[18, 36, 54, 72, 90] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(4, 7, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 7, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0068/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 69. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[19, 38, 57, 76, 95, 114] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(5, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0069/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 70. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[20, 40, 60, 80, 100, 120, 140] for f(t)=sum coefficients[k]*t^k and derivative order=8≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(6, 8, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 8, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0070/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 71. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[21, 42, 63, 84] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(3, 6, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 6, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0071/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 72. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[22, 44, 66, 88, 110] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(4, 5, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 5, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0072/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 73. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[23, 46, 69, 92, 115, 138] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(5, 7, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 7, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0073/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 74. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[24, 48, 72, 96, 120, 144, 168] for f(t)=sum coefficients[k]*t^k and derivative order=9≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(6, 9, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 9, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0074/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 75. Orders above the Declared Polynomial Degree Bound Give Zero

Use original low-to-high coefficients=[25, 50, 75, 100] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).

View answer and explanation(3, 4, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 4, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0075/)

Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate





## 76. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=1 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(1, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (1, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0076/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 77. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=2 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(2, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (2, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0077/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 78. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=3 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(3, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (3, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0078/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 79. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=4 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(4, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (4, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0079/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 80. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=5 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(5, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (5, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0080/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 81. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=6 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(6, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (6, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0081/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 82. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=7 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(7, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (7, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0082/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 83. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=8 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(8, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (8, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0083/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 84. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=9 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(9, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (9, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0084/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 85. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=10 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(10, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (10, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0085/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 86. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=11 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(11, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (11, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0086/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 87. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=12 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(12, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (12, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0087/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 88. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=13 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(13, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (13, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0088/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 89. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=14 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(14, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (14, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0089/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 90. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=15 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(15, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (15, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0090/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 91. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=16 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(16, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (16, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0091/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 92. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=17 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(17, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (17, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0092/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 93. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=18 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(18, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (18, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0093/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 94. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=19 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(19, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (19, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0094/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 95. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=20 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(20, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (20, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0095/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 96. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=21 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(21, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (21, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0096/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 97. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=22 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(22, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (22, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0097/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 98. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=23 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(23, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (23, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0098/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 99. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=24 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(24, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (24, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0099/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate





## 100. A First Derivative May Exist Where the Second Does Not

Use nonzero scale=25 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).

View answer and explanation(25, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)

The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (25, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0100/)

Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261111-r02-higher-order-derivative-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261111-r02-higher-order-derivative-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261111-r02-higher-order-derivative-foundations/validation-report.json)