# Polynomial Differentiation Rule Foundations

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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-20261007-advance-20261008-r05-derivative-rules-foundations · 2026-10-07 · 100 questions

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



## 1. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [3, 1], starting with its constant term; g has ascending coefficients [-5, -1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation0

Use f′+g′. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0001/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 2. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [-11, 2, 0, 2], starting with its constant term; g has ascending coefficients [1, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation8

Use f′+g′. The exact result is 8; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0002/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 3. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [-10, 3, 1, 3], starting with its constant term; g has ascending coefficients [0, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation2

Use f′+g′. The exact result is 2; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0003/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 4. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term; g has ascending coefficients [-1, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation18

Use f′+g′. The exact result is 18; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0004/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 5. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term; g has ascending coefficients [-2, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation8

Use f′+g′. The exact result is 8; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0005/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 6. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [-7, 6, 1, 2], starting with its constant term; g has ascending coefficients [-3, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation33

Use f′+g′. The exact result is 33; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0006/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 7. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [-6, 7, -1, 3], starting with its constant term; g has ascending coefficients [-4, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation12

Use f′+g′. The exact result is 12; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0007/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 8. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [-5, 8, 0, 4], starting with its constant term; g has ascending coefficients [-5, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation7

Use f′+g′. The exact result is 7; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0008/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 9. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term; g has ascending coefficients [-6, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation12

Use f′+g′. The exact result is 12; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0009/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 10. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term; g has ascending coefficients [-7, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation45

Use f′+g′. The exact result is 45; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0010/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 11. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term; g has ascending coefficients [-8, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation30

Use f′+g′. The exact result is 30; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0011/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 12. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [-1, 12, 1, 4], starting with its constant term; g has ascending coefficients [-9, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation22

Use f′+g′. The exact result is 22; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0012/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 13. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term; g has ascending coefficients [-10, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation12

Use f′+g′. The exact result is 12; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0013/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 14. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term; g has ascending coefficients [-11, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation24

Use f′+g′. The exact result is 24; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0014/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 15. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [2, 15, 1, 3], starting with its constant term; g has ascending coefficients [-12, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation46

Use f′+g′. The exact result is 46; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0015/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 16. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [3, 16, -1, 4], starting with its constant term; g has ascending coefficients [-13, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation75

Use f′+g′. The exact result is 75; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0016/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 17. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [4, 17, 0, 1], starting with its constant term; g has ascending coefficients [-14, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation14

Use f′+g′. The exact result is 14; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0017/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 18. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term; g has ascending coefficients [-15, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation17

Use f′+g′. The exact result is 17; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0018/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 19. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term; g has ascending coefficients [-16, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation24

Use f′+g′. The exact result is 24; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0019/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 20. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [7, 20, 0, 4], starting with its constant term; g has ascending coefficients [-17, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation83

Use f′+g′. The exact result is 83; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0020/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 21. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term; g has ascending coefficients [-18, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation12

Use f′+g′. The exact result is 12; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0021/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 22. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [9, 22, -1, 2], starting with its constant term; g has ascending coefficients [-19, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation30

Use f′+g′. The exact result is 30; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0022/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 23. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [10, 23, 0, 3], starting with its constant term; g has ascending coefficients [-20, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation22

Use f′+g′. The exact result is 22; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0023/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 24. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term; g has ascending coefficients [-21, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation42

Use f′+g′. The exact result is 42; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0024/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 25. Derivative of a Polynomial Sum

On all R, f has ascending coefficients [12, 25, -1, 1], starting with its constant term; g has ascending coefficients [-22, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation24

Use f′+g′. The exact result is 24; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0025/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate





## 26. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [1, 1], starting with its constant term. Find the derivative of (0)f at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation0

Use c*f′, including c=0. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0026/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 27. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [-11, 2, 0, 2], starting with its constant term. Find the derivative of (-2)f at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-16

Use c*f′, including c=0. The exact result is -16; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0027/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 28. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [-10, 3, 1, 3], starting with its constant term. Find the derivative of (3)f at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation9

Use c*f′, including c=0. The exact result is 9; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0028/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 29. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term. Find the derivative of (-4)f at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-56

Use c*f′, including c=0. The exact result is -56; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0029/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 30. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term. Find the derivative of (1)f at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation17

Use c*f′, including c=0. The exact result is 17; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0030/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 31. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [-7, 6, 1, 2], starting with its constant term. Find the derivative of (-2)f at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-52

Use c*f′, including c=0. The exact result is -52; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0031/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 32. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [-6, 7, -1, 3], starting with its constant term. Find the derivative of (3)f at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation54

Use c*f′, including c=0. The exact result is 54; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0032/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 33. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [-5, 8, 0, 4], starting with its constant term. Find the derivative of (-4)f at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-32

Use c*f′, including c=0. The exact result is -32; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0033/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 34. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term. Find the derivative of (1)f at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation14

Use c*f′, including c=0. The exact result is 14; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0034/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 35. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term. Find the derivative of (-2)f at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-60

Use c*f′, including c=0. The exact result is -60; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0035/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 36. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term. Find the derivative of (3)f at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation141

Use c*f′, including c=0. The exact result is 141; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0036/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 37. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [-1, 12, 1, 4], starting with its constant term. Find the derivative of (-4)f at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-88

Use c*f′, including c=0. The exact result is -88; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0037/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 38. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term. Find the derivative of (1)f at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation13

Use c*f′, including c=0. The exact result is 13; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0038/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 39. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term. Find the derivative of (-2)f at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-40

Use c*f′, including c=0. The exact result is -40; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0039/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 40. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [2, 15, 1, 3], starting with its constant term. Find the derivative of (3)f at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation165

Use c*f′, including c=0. The exact result is 165; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0040/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 41. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [3, 16, -1, 4], starting with its constant term. Find the derivative of (-4)f at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-272

Use c*f′, including c=0. The exact result is -272; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0041/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 42. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [4, 17, 0, 1], starting with its constant term. Find the derivative of (1)f at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation20

Use c*f′, including c=0. The exact result is 20; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0042/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 43. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term. Find the derivative of (-2)f at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-36

Use c*f′, including c=0. The exact result is -36; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0043/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 44. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term. Find the derivative of (3)f at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation78

Use c*f′, including c=0. The exact result is 78; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0044/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 45. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [7, 20, 0, 4], starting with its constant term. Find the derivative of (-4)f at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-272

Use c*f′, including c=0. The exact result is -272; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0045/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 46. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term. Find the derivative of (1)f at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation29

Use c*f′, including c=0. The exact result is 29; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0046/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 47. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [9, 22, -1, 2], starting with its constant term. Find the derivative of (-2)f at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-60

Use c*f′, including c=0. The exact result is -60; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0047/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 48. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [10, 23, 0, 3], starting with its constant term. Find the derivative of (3)f at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation69

Use c*f′, including c=0. The exact result is 69; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0048/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 49. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term. Find the derivative of (-4)f at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-152

Use c*f′, including c=0. The exact result is -152; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0049/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 50. Derivative of a Scalar Multiple of a Polynomial

On all R, f has ascending coefficients [12, 25, -1, 1], starting with its constant term. Find the derivative of (1)f at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation33

Use c*f′, including c=0. The exact result is 33; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0050/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate





## 51. Derivative of a Polynomial Product

On all R, f has ascending coefficients [3], starting with its constant term; g has ascending coefficients [-2], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation0

Use f′g+fg′, not f′g′. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0051/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 52. Derivative of a Polynomial Product

On all R, f has ascending coefficients [-11, 2, 0, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 3], starting with its constant term. Find the derivative of fg at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-105

Use f′g+fg′, not f′g′. The exact result is -105; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0052/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 53. Derivative of a Polynomial Product

On all R, f has ascending coefficients [-10, 3, 1, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 4], starting with its constant term. Find the derivative of fg at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-4

Use f′g+fg′, not f′g′. The exact result is -4; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0053/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 54. Derivative of a Polynomial Product

On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 2], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation78

Use f′g+fg′, not f′g′. The exact result is 78; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0054/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 55. Derivative of a Polynomial Product

On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 3], starting with its constant term. Find the derivative of fg at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation866

Use f′g+fg′, not f′g′. The exact result is 866; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0055/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 56. Derivative of a Polynomial Product

On all R, f has ascending coefficients [-7, 6, 1, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 4], starting with its constant term. Find the derivative of fg at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-2040

Use f′g+fg′, not f′g′. The exact result is -2040; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0056/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 57. Derivative of a Polynomial Product

On all R, f has ascending coefficients [-6, 7, -1, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 2], starting with its constant term. Find the derivative of fg at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-85

Use f′g+fg′, not f′g′. The exact result is -85; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0057/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 58. Derivative of a Polynomial Product

On all R, f has ascending coefficients [-5, 8, 0, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 3], starting with its constant term. Find the derivative of fg at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation6

Use f′g+fg′, not f′g′. The exact result is 6; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0058/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 59. Derivative of a Polynomial Product

On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 4], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation175

Use f′g+fg′, not f′g′. The exact result is 175; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0059/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 60. Derivative of a Polynomial Product

On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 2], starting with its constant term. Find the derivative of fg at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation1472

Use f′g+fg′, not f′g′. The exact result is 1472; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0060/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 61. Derivative of a Polynomial Product

On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 3], starting with its constant term. Find the derivative of fg at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-2524

Use f′g+fg′, not f′g′. The exact result is -2524; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0061/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 62. Derivative of a Polynomial Product

On all R, f has ascending coefficients [-1, 12, 1, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 4], starting with its constant term. Find the derivative of fg at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-258

Use f′g+fg′, not f′g′. The exact result is -258; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0062/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 63. Derivative of a Polynomial Product

On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 2], starting with its constant term. Find the derivative of fg at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation26

Use f′g+fg′, not f′g′. The exact result is 26; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0063/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 64. Derivative of a Polynomial Product

On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 3], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation307

Use f′g+fg′, not f′g′. The exact result is 307; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0064/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 65. Derivative of a Polynomial Product

On all R, f has ascending coefficients [2, 15, 1, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 4], starting with its constant term. Find the derivative of fg at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation5380

Use f′g+fg′, not f′g′. The exact result is 5380; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0065/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 66. Derivative of a Polynomial Product

On all R, f has ascending coefficients [3, 16, -1, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 2], starting with its constant term. Find the derivative of fg at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-2382

Use f′g+fg′, not f′g′. The exact result is -2382; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0066/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 67. Derivative of a Polynomial Product

On all R, f has ascending coefficients [4, 17, 0, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 3], starting with its constant term. Find the derivative of fg at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-64

Use f′g+fg′, not f′g′. The exact result is -64; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0067/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 68. Derivative of a Polynomial Product

On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 4], starting with its constant term. Find the derivative of fg at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation36

Use f′g+fg′, not f′g′. The exact result is 36; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0068/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 69. Derivative of a Polynomial Product

On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 2], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation399

Use f′g+fg′, not f′g′. The exact result is 399; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0069/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 70. Derivative of a Polynomial Product

On all R, f has ascending coefficients [7, 20, 0, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 3], starting with its constant term. Find the derivative of fg at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation5630

Use f′g+fg′, not f′g′. The exact result is 5630; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0070/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 71. Derivative of a Polynomial Product

On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 4], starting with its constant term. Find the derivative of fg at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-2330

Use f′g+fg′, not f′g′. The exact result is -2330; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0071/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 72. Derivative of a Polynomial Product

On all R, f has ascending coefficients [9, 22, -1, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 2], starting with its constant term. Find the derivative of fg at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-34

Use f′g+fg′, not f′g′. The exact result is -34; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0072/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 73. Derivative of a Polynomial Product

On all R, f has ascending coefficients [10, 23, 0, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 3], starting with its constant term. Find the derivative of fg at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation56

Use f′g+fg′, not f′g′. The exact result is 56; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0073/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 74. Derivative of a Polynomial Product

On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 4], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation982

Use f′g+fg′, not f′g′. The exact result is 982; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0074/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 75. Derivative of a Polynomial Product

On all R, f has ascending coefficients [12, 25, -1, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 2], starting with its constant term. Find the derivative of fg at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation2442

Use f′g+fg′, not f′g′. The exact result is 2442; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0075/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate





## 76. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [1, 2, 3], starting with its constant term; g has ascending coefficients [4], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation0

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0076/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 77. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [-11, 2, 1, 1], starting with its constant term; g has ascending coefficients [-4, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation42

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 42; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0077/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 78. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [-10, 3, 2, 1], starting with its constant term; g has ascending coefficients [-3, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation230

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 230; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0078/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 79. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [-9, 4, 0, 1], starting with its constant term; g has ascending coefficients [-2, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation7412

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 7412; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0079/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 80. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [-8, 1, 1, 1], starting with its constant term; g has ascending coefficients [-1, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation0

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0080/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 81. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [-7, 2, 2, 1], starting with its constant term; g has ascending coefficients [0, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation2

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 2; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0081/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 82. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [-6, 3, 0, 1], starting with its constant term; g has ascending coefficients [1, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation306

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 306; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0082/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 83. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [-5, 4, 1, 1], starting with its constant term; g has ascending coefficients [2, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation26124

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 26124; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0083/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 84. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [-4, 1, 2, 1], starting with its constant term; g has ascending coefficients [3, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-130

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is -130; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0084/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 85. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [-3, 2, 0, 1], starting with its constant term; g has ascending coefficients [4, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation50

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 50; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0085/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 86. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [-2, 3, 1, 1], starting with its constant term; g has ascending coefficients [5, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation2112

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 2112; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0086/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 87. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [-1, 4, 2, 1], starting with its constant term; g has ascending coefficients [6, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation61700

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 61700; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0087/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 88. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [0, 1, 0, 1], starting with its constant term; g has ascending coefficients [7, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation218

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 218; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0088/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 89. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [1, 2, 1, 1], starting with its constant term; g has ascending coefficients [8, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation210

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 210; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0089/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 90. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [2, 3, 2, 1], starting with its constant term; g has ascending coefficients [9, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation6470

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 6470; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0090/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 91. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [3, 4, 0, 1], starting with its constant term; g has ascending coefficients [10, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation29444

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 29444; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0091/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 92. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [4, 1, 1, 1], starting with its constant term; g has ascending coefficients [11, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation0

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0092/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 93. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [5, 2, 2, 1], starting with its constant term; g has ascending coefficients [12, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation482

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 482; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0093/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 94. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [6, 3, 0, 1], starting with its constant term; g has ascending coefficients [13, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation4626

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 4626; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0094/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 95. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [7, 4, 1, 1], starting with its constant term; g has ascending coefficients [14, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation65940

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 65940; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0095/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 96. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [8, 1, 2, 1], starting with its constant term; g has ascending coefficients [15, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation-1666

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is -1666; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0096/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 97. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [9, 2, 0, 1], starting with its constant term; g has ascending coefficients [16, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation770

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 770; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0097/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 98. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [10, 3, 1, 1], starting with its constant term; g has ascending coefficients [17, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation10944

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 10944; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0098/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 99. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [11, 4, 2, 1], starting with its constant term; g has ascending coefficients [18, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation124100

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 124100; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0099/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate





## 100. Chain-Rule Derivative of a Polynomial Composition

On all R, f has ascending coefficients [12, 1, 0, 1], starting with its constant term; g has ascending coefficients [19, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.

View answer and explanation1946

Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 1946; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r05-derivative-rules-0100/)

Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261007-advance-20261008-r05-derivative-rules-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261007-advance-20261008-r05-derivative-rules-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261007-advance-20261008-r05-derivative-rules-foundations/validation-report.json)