# Question 23033: Order Zero Is the Original Function; Later Orders Need the Recursive Chain [Return to catalogue page 116](https://ocme.evemisslab.com/en/questions/page/116/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261111-r02-higher-order-derivative-foundations/#q-20261014-advance-20261111-r02-higher-order-derivative-0008) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0045-23033 · q-20261014-advance-20261111-r02-higher-order-derivative-0008 ## Original question and givens 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). - G001 — 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). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-higher-order-derivative](https://ocme.evemisslab.com/en/mko/mko-higher-order-derivative/) ## Step-by-step derivation S001 · impliesBefore source operation=higher-order-qualification;original snapshot and mathematics retained After explicit ordered source output fields=[declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate] Conditions: none Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL low-to-high coefficients=[8, 16, 24, 32, 40, 48, 56];declared order=7 After declared degree BOUND=6=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list Conditions: none Length minus1 is the declared bound;do not trim trailing zeros.Order0 is the original function.The monomial rule and finite sums justify each polynomial derivative,not an infinite extrapolation for arbitrary functions. Step identifiers and dependenciesRule: higher_ORIGINAL_low_to_high_coefficients_degree_BOUND_order0 · Uses: G001, S001 S003 · calculationBefore Higher derivative level1 ORIGINAL power1 coefficient product := mul(16, 1) After Higher derivative level1 ORIGINAL power1 coefficient product = 16 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Higher derivative level1 ORIGINAL power2 coefficient product := mul(24, 2) After Higher derivative level1 ORIGINAL power2 coefficient product = 48 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Higher derivative level1 ORIGINAL power3 coefficient product := mul(32, 3) After Higher derivative level1 ORIGINAL power3 coefficient product = 96 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Higher derivative level1 ORIGINAL power4 coefficient product := mul(40, 4) After Higher derivative level1 ORIGINAL power4 coefficient product = 160 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Higher derivative level1 ORIGINAL power5 coefficient product := mul(48, 5) After Higher derivative level1 ORIGINAL power5 coefficient product = 240 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Higher derivative level1 ORIGINAL power6 coefficient product := mul(56, 6) After Higher derivative level1 ORIGINAL power6 coefficient product = 336 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore previous derivative order=0;actual low-to-high list=[8, 16, 24, 32, 40, 48, 56];drop ONLY its constant term After ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 16, 1, 16], [2, 24, 2, 48], [3, 32, 3, 96], [4, 40, 4, 160], [5, 48, 5, 240], [6, 56, 6, 336]];next raw list=[16, 48, 96, 160, 240, 336];all intermediate products SAFE Conditions: none At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value. Step identifiers and dependenciesRule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S008 S010 · calculationBefore Higher derivative level2 ORIGINAL power1 coefficient product := mul(48, 1) After Higher derivative level2 ORIGINAL power1 coefficient product = 48 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Higher derivative level2 ORIGINAL power2 coefficient product := mul(96, 2) After Higher derivative level2 ORIGINAL power2 coefficient product = 192 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Higher derivative level2 ORIGINAL power3 coefficient product := mul(160, 3) After Higher derivative level2 ORIGINAL power3 coefficient product = 480 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Higher derivative level2 ORIGINAL power4 coefficient product := mul(240, 4) After Higher derivative level2 ORIGINAL power4 coefficient product = 960 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Higher derivative level2 ORIGINAL power5 coefficient product := mul(336, 5) After Higher derivative level2 ORIGINAL power5 coefficient product = 1680 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore previous derivative order=1;actual low-to-high list=[16, 48, 96, 160, 240, 336];drop ONLY its constant term After ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 48, 1, 48], [2, 96, 2, 192], [3, 160, 3, 480], [4, 240, 4, 960], [5, 336, 5, 1680]];next raw list=[48, 192, 480, 960, 1680];all intermediate products SAFE Conditions: none At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value. Step identifiers and dependenciesRule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S014 S016 · calculationBefore Higher derivative level3 ORIGINAL power1 coefficient product := mul(192, 1) After Higher derivative level3 ORIGINAL power1 coefficient product = 192 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Higher derivative level3 ORIGINAL power2 coefficient product := mul(480, 2) After Higher derivative level3 ORIGINAL power2 coefficient product = 960 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Higher derivative level3 ORIGINAL power3 coefficient product := mul(960, 3) After Higher derivative level3 ORIGINAL power3 coefficient product = 2880 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Higher derivative level3 ORIGINAL power4 coefficient product := mul(1680, 4) After Higher derivative level3 ORIGINAL power4 coefficient product = 6720 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore previous derivative order=2;actual low-to-high list=[48, 192, 480, 960, 1680];drop ONLY its constant term After ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 192, 1, 192], [2, 480, 2, 960], [3, 960, 3, 2880], [4, 1680, 4, 6720]];next raw list=[192, 960, 2880, 6720];all intermediate products SAFE Conditions: none At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value. Step identifiers and dependenciesRule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S019 S021 · calculationBefore Higher derivative level4 ORIGINAL power1 coefficient product := mul(960, 1) After Higher derivative level4 ORIGINAL power1 coefficient product = 960 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Higher derivative level4 ORIGINAL power2 coefficient product := mul(2880, 2) After Higher derivative level4 ORIGINAL power2 coefficient product = 5760 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Higher derivative level4 ORIGINAL power3 coefficient product := mul(6720, 3) After Higher derivative level4 ORIGINAL power3 coefficient product = 20160 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore previous derivative order=3;actual low-to-high list=[192, 960, 2880, 6720];drop ONLY its constant term After ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 960, 1, 960], [2, 2880, 2, 5760], [3, 6720, 3, 20160]];next raw list=[960, 5760, 20160];all intermediate products SAFE Conditions: none At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value. Step identifiers and dependenciesRule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S023 S025 · calculationBefore Higher derivative level5 ORIGINAL power1 coefficient product := mul(5760, 1) After Higher derivative level5 ORIGINAL power1 coefficient product = 5760 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore Higher derivative level5 ORIGINAL power2 coefficient product := mul(20160, 2) After Higher derivative level5 ORIGINAL power2 coefficient product = 40320 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · impliesBefore previous derivative order=4;actual low-to-high list=[960, 5760, 20160];drop ONLY its constant term After ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 5760, 1, 5760], [2, 20160, 2, 40320]];next raw list=[5760, 40320];all intermediate products SAFE Conditions: none At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value. Step identifiers and dependenciesRule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S026 S028 · calculationBefore Higher derivative level6 ORIGINAL power1 coefficient product := mul(40320, 1) After Higher derivative level6 ORIGINAL power1 coefficient product = 40320 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · impliesBefore previous derivative order=5;actual low-to-high list=[5760, 40320];drop ONLY its constant term After ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 40320, 1, 40320]];next raw list=[40320];all intermediate products SAFE Conditions: none At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value. Step identifiers and dependenciesRule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S028 S030 · impliesBefore previous derivative order=6;actual low-to-high list=[40320];drop ONLY its constant term After ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[];next raw list=[];all intermediate products SAFE Conditions: none At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value. Step identifiers and dependenciesRule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S029 S031 · impliesBefore ORIGINAL coefficients=[8, 16, 24, 32, 40, 48, 56];order=7;declared degree bound=6 After final derivative coefficients=[0];for each original power k>=order multiplier=k*(k-1)*...*(k-order+1);lower powers vanish Conditions: none Repeated differentiation gives the falling factorial in original coefficient order;orders above the declared bound vanish.This gives no analyticity claim for arbitrary differentiable functions and does not turn a trailing-zero bound into the minimal degree. Step identifiers and dependenciesRule: higher_falling_factorial_rule_all_orders_polynomial_only · Uses: G001, S030 ## Reverse checks - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Recompute Higher derivative level6 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Recompute Higher derivative level5 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Higher derivative level5 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Recompute Higher derivative level4 ORIGINAL power3 coefficient product with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Higher derivative level4 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Higher derivative level4 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Recompute Higher derivative level3 ORIGINAL power4 coefficient product with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Higher derivative level3 ORIGINAL power3 coefficient product with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Higher derivative level3 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Higher derivative level3 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Recompute Higher derivative level2 ORIGINAL power5 coefficient product with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Higher derivative level2 ORIGINAL power4 coefficient product with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Higher derivative level2 ORIGINAL power3 coefficient product with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Higher derivative level2 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Higher derivative level2 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute Higher derivative level1 ORIGINAL power6 coefficient product with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Higher derivative level1 ORIGINAL power5 coefficient product with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Higher derivative level1 ORIGINAL power4 coefficient product with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Higher derivative level1 ORIGINAL power3 coefficient product with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Higher derivative level1 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Higher derivative level1 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (6, 7, Yes, recursive_derivative_chain_required, Yes) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: d6775aba27fcdfe822d74aaae61f24f18bef1222dca0939f8e064ec662704593 - candidate record SHA-256: 1e4c582ae987ecdf16c3b81c942ad1492aafa635b6134e4487a926e5416782e4 - semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438 - question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0008.json)