# Polynomial Differentiation Order and Coefficient Depth

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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-20261009-r04-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. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [0], constant term first. Find f″(2). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation0/1

Differentiate twice and evaluate exactly, obtaining 0/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0001/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 2. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [-11, 2, 2, 0], constant term first. Find f″(-1). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation4/1

Differentiate twice and evaluate exactly, obtaining 4/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0002/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 3. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [-10, 3, 3, 1], constant term first. Find f″(0). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation6/1

Differentiate twice and evaluate exactly, obtaining 6/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0003/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 4. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [-9, 1, 4, -1], constant term first. Find f″(1). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation2/1

Differentiate twice and evaluate exactly, obtaining 2/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0004/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 5. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [-8, 2, 5, 0], constant term first. Find f″(2). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation10/1

Differentiate twice and evaluate exactly, obtaining 10/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0005/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 6. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [-7, 3, 6, 1], constant term first. Find f″(-2). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation0/1

Differentiate twice and evaluate exactly, obtaining 0/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0006/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 7. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [-6, 1, 7, -1], constant term first. Find f″(-1). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation20/1

Differentiate twice and evaluate exactly, obtaining 20/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0007/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 8. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [-5, 2, 8, 0], constant term first. Find f″(0). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation16/1

Differentiate twice and evaluate exactly, obtaining 16/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0008/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 9. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [-4, 3, 9, 1], constant term first. Find f″(1). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation24/1

Differentiate twice and evaluate exactly, obtaining 24/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0009/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 10. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [-3, 1, 10, -1], constant term first. Find f″(2). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation8/1

Differentiate twice and evaluate exactly, obtaining 8/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0010/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 11. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [-2, 2, 11, 0], constant term first. Find f″(-2). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation22/1

Differentiate twice and evaluate exactly, obtaining 22/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0011/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 12. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [-1, 3, 12, 1], constant term first. Find f″(-1). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation18/1

Differentiate twice and evaluate exactly, obtaining 18/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0012/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 13. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [0, 1, 13, -1], constant term first. Find f″(0). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation26/1

Differentiate twice and evaluate exactly, obtaining 26/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0013/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 14. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [1, 2, 14, 0], constant term first. Find f″(1). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation28/1

Differentiate twice and evaluate exactly, obtaining 28/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0014/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 15. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [2, 3, 15, 1], constant term first. Find f″(2). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation42/1

Differentiate twice and evaluate exactly, obtaining 42/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0015/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 16. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [3, 1, 16, -1], constant term first. Find f″(-2). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation44/1

Differentiate twice and evaluate exactly, obtaining 44/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0016/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 17. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [4, 2, 17, 0], constant term first. Find f″(-1). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation34/1

Differentiate twice and evaluate exactly, obtaining 34/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0017/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 18. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [5, 3, 18, 1], constant term first. Find f″(0). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation36/1

Differentiate twice and evaluate exactly, obtaining 36/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0018/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 19. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [6, 1, 19, -1], constant term first. Find f″(1). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation32/1

Differentiate twice and evaluate exactly, obtaining 32/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0019/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 20. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [7, 2, 20, 0], constant term first. Find f″(2). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation40/1

Differentiate twice and evaluate exactly, obtaining 40/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0020/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 21. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [8, 3, 21, 1], constant term first. Find f″(-2). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation30/1

Differentiate twice and evaluate exactly, obtaining 30/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0021/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 22. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [9, 1, 22, -1], constant term first. Find f″(-1). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation50/1

Differentiate twice and evaluate exactly, obtaining 50/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0022/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 23. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [10, 2, 23, 0], constant term first. Find f″(0). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation46/1

Differentiate twice and evaluate exactly, obtaining 46/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0023/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 24. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [11, 3, 24, 1], constant term first. Find f″(1). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation54/1

Differentiate twice and evaluate exactly, obtaining 54/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0024/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 25. Second Derivative of a Polynomial at a Point

On all R, f has ascending coefficients [12, 1, 25, -1], constant term first. Find f″(2). Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation38/1

Differentiate twice and evaluate exactly, obtaining 38/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0025/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-second-value · candidate





## 26. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [0], constant term first and g has ascending coefficients [2, 1], constant term first. At x=3, is 0 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0026/)

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





## 27. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [5], constant term first and g has ascending coefficients [-2], constant term first. At x=1, is 1 the derivative of fg?

View answer and explanationNo

Compare the candidate with the exact derivative of the complete product, giving No. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0027/)

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





## 28. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [-10, 3, 1, 3], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=1, is 40 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0028/)

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





## 29. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [-9, 4, -1, 4], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=2, is 981 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0029/)

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





## 30. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [-8, 5, 0, 1], constant term first and g has ascending coefficients [2, -1, 1, 2], constant term first. At x=-1, is -25 the derivative of fg?

View answer and explanationNo

Compare the candidate with the exact derivative of the complete product, giving No. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0030/)

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





## 31. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [-7, 6, 1, 2], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=0, is 19 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0031/)

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





## 32. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [-6, 7, -1, 3], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=1, is 54 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0032/)

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





## 33. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [-5, 8, 0, 4], constant term first and g has ascending coefficients [2, -1, 1, 2], constant term first. At x=2, is 2282 the derivative of fg?

View answer and explanationNo

Compare the candidate with the exact derivative of the complete product, giving No. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0033/)

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





## 34. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [-4, 9, 1, 1], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=-1, is -68 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0034/)

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





## 35. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [-3, 10, -1, 2], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=0, is 23 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0035/)

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





## 36. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [-2, 11, 0, 3], constant term first and g has ascending coefficients [2, -1, 1, 2], constant term first. At x=1, is 165 the derivative of fg?

View answer and explanationNo

Compare the candidate with the exact derivative of the complete product, giving No. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0036/)

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





## 37. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [-1, 12, 1, 4], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=2, is 4093 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0037/)

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





## 38. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [0, 13, -1, 1], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=-1, is 54 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0038/)

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





## 39. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [1, 14, 0, 2], constant term first and g has ascending coefficients [2, -1, 1, 2], constant term first. At x=0, is 28 the derivative of fg?

View answer and explanationNo

Compare the candidate with the exact derivative of the complete product, giving No. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0039/)

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





## 40. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [2, 15, 1, 3], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=1, is 340 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0040/)

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





## 41. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [3, 16, -1, 4], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=2, is 1665 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0041/)

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





## 42. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [4, 17, 0, 1], constant term first and g has ascending coefficients [2, -1, 1, 2], constant term first. At x=-1, is -1 the derivative of fg?

View answer and explanationNo

Compare the candidate with the exact derivative of the complete product, giving No. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0042/)

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





## 43. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [5, 18, 1, 2], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=0, is 31 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0043/)

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





## 44. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [6, 19, -1, 3], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=1, is 186 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0044/)

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





## 45. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [7, 20, 0, 4], constant term first and g has ascending coefficients [2, -1, 1, 2], constant term first. At x=2, is 3494 the derivative of fg?

View answer and explanationNo

Compare the candidate with the exact derivative of the complete product, giving No. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0045/)

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





## 46. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [8, 21, 1, 1], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=-1, is -56 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0046/)

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





## 47. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [9, 22, -1, 2], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=0, is 35 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0047/)

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





## 48. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [10, 23, 0, 3], constant term first and g has ascending coefficients [2, -1, 1, 2], constant term first. At x=1, is 381 the derivative of fg?

View answer and explanationNo

Compare the candidate with the exact derivative of the complete product, giving No. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0048/)

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





## 49. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [11, 24, 1, 4], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=2, is 5833 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0049/)

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





## 50. Eligibility of a Candidate Product-Derivative Value

On all R, f has ascending coefficients [12, 25, -1, 1], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=-1, is 90 the derivative of fg?

View answer and explanationYes

Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0050/)

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





## 51. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [3], constant term first and g has ascending coefficients [1, 1], constant term first. At x=2, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(0/1, 0/1)

Expand and differentiate each ordered composition without truncation to obtain (0/1, 0/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0051/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 52. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [-11, 2, 0, 2], constant term first and g has ascending coefficients [-4, 1, 2, 1], constant term first. At x=0, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(98/1, 640/1)

Expand and differentiate each ordered composition without truncation to obtain (98/1, 640/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0052/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 53. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [-10, 3, 1, 3], constant term first and g has ascending coefficients [-3, 1, 3, 1], constant term first. At x=1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(430/1, 140/1)

Expand and differentiate each ordered composition without truncation to obtain (430/1, 140/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0053/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 54. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [-9, 4, -1, 4], constant term first and g has ascending coefficients [-2, 1, 1, 1], constant term first. At x=2, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(29036/1, 107616/1)

Expand and differentiate each ordered composition without truncation to obtain (29036/1, 107616/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0054/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 55. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [-8, 5, 0, 1], constant term first and g has ascending coefficients [-1, 1, 2, 1], constant term first. At x=-1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(0/1, 4264/1)

Expand and differentiate each ordered composition without truncation to obtain (0/1, 4264/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0055/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 56. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [-7, 6, 1, 2], constant term first and g has ascending coefficients [0, 1, 3, 1], constant term first. At x=0, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(6/1, 636/1)

Expand and differentiate each ordered composition without truncation to obtain (6/1, 636/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0056/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 57. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [-6, 7, -1, 3], constant term first and g has ascending coefficients [1, 1, 1, 1], constant term first. At x=1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(858/1, 476/1)

Expand and differentiate each ordered composition without truncation to obtain (858/1, 476/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0057/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 58. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [-5, 8, 0, 4], constant term first and g has ascending coefficients [2, 1, 2, 1], constant term first. At x=2, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(100968/1, 320320/1)

Expand and differentiate each ordered composition without truncation to obtain (100968/1, 320320/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0058/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 59. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [-4, 9, 1, 1], constant term first and g has ascending coefficients [3, 1, 3, 1], constant term first. At x=-1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(-130/1, 4300/1)

Expand and differentiate each ordered composition without truncation to obtain (-130/1, 4300/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0059/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 60. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [-3, 10, -1, 2], constant term first and g has ascending coefficients [4, 1, 1, 1], constant term first. At x=0, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(98/1, 220/1)

Expand and differentiate each ordered composition without truncation to obtain (98/1, 220/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0060/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 61. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [-2, 11, 0, 3], constant term first and g has ascending coefficients [5, 1, 2, 1], constant term first. At x=1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(5920/1, 9620/1)

Expand and differentiate each ordered composition without truncation to obtain (5920/1, 9620/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0061/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 62. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [-1, 12, 1, 4], constant term first and g has ascending coefficients [6, 1, 3, 1], constant term first. At x=2, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(236900/1, 691072/1)

Expand and differentiate each ordered composition without truncation to obtain (236900/1, 691072/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0062/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 63. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [0, 13, -1, 1], constant term first and g has ascending coefficients [7, 1, 1, 1], constant term first. At x=-1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(218/1, 11628/1)

Expand and differentiate each ordered composition without truncation to obtain (218/1, 11628/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0063/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 64. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [1, 14, 0, 2], constant term first and g has ascending coefficients [8, 1, 2, 1], constant term first. At x=0, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(398/1, 112/1)

Expand and differentiate each ordered composition without truncation to obtain (398/1, 112/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0064/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 65. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [2, 15, 1, 3], constant term first and g has ascending coefficients [9, 1, 3, 1], constant term first. At x=1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(18070/1, 37700/1)

Expand and differentiate each ordered composition without truncation to obtain (18070/1, 37700/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0065/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 66. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [3, 16, -1, 4], constant term first and g has ascending coefficients [10, 1, 1, 1], constant term first. At x=2, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(116960/1, 722040/1)

Expand and differentiate each ordered composition without truncation to obtain (116960/1, 722040/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0066/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 67. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [4, 17, 0, 1], constant term first and g has ascending coefficients [11, 1, 2, 1], constant term first. At x=-1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(0/1, 10660/1)

Expand and differentiate each ordered composition without truncation to obtain (0/1, 10660/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0067/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 68. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [5, 18, 1, 2], constant term first and g has ascending coefficients [12, 1, 3, 1], constant term first. At x=0, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(906/1, 1908/1)

Expand and differentiate each ordered composition without truncation to obtain (906/1, 1908/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0068/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 69. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [6, 19, -1, 3], constant term first and g has ascending coefficients [13, 1, 1, 1], constant term first. At x=1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(13746/1, 58292/1)

Expand and differentiate each ordered composition without truncation to obtain (13746/1, 58292/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0069/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 70. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [7, 20, 0, 4], constant term first and g has ascending coefficients [14, 1, 2, 1], constant term first. At x=2, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(258468/1, 1294720/1)

Expand and differentiate each ordered composition without truncation to obtain (258468/1, 1294720/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0070/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 71. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [8, 21, 1, 1], constant term first and g has ascending coefficients [15, 1, 3, 1], constant term first. At x=-1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(-1642/1, 9460/1)

Expand and differentiate each ordered composition without truncation to obtain (-1642/1, 9460/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0071/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 72. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [9, 22, -1, 2], constant term first and g has ascending coefficients [16, 1, 1, 1], constant term first. At x=0, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(1526/1, 5764/1)

Expand and differentiate each ordered composition without truncation to obtain (1526/1, 5764/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0072/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 73. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [10, 23, 0, 3], constant term first and g has ascending coefficients [17, 1, 2, 1], constant term first. At x=1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(31936/1, 129056/1)

Expand and differentiate each ordered composition without truncation to obtain (31936/1, 129056/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0073/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 74. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [11, 24, 1, 4], constant term first and g has ascending coefficients [18, 1, 3, 1], constant term first. At x=2, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(482600/1, 2101096/1)

Expand and differentiate each ordered composition without truncation to obtain (482600/1, 2101096/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0074/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 75. Derivatives in Both Composition Orders

On all R, f has ascending coefficients [12, 25, -1, 1], constant term first and g has ascending coefficients [19, 1, 1, 1], constant term first. At x=-1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation(1922/1, 19380/1)

Expand and differentiate each ordered composition without truncation to obtain (1922/1, 19380/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0075/)

Differentiating polynomial sums, products, and compositions · qf-density-derivative-chain-order-pair · candidate





## 76. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [2, 1], constant term first and g has ascending coefficients [-4, -1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[0]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [0]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0076/)

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





## 77. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 1, 3, 2], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[2, 8, 9]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [2, 8, 9]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0077/)

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





## 78. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 2, 4, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[3, 10, 12]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [3, 10, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0078/)

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





## 79. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 3, 5, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[4, 12, 6]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [4, 12, 6]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0079/)

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





## 80. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 4, 6, 2], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[5, 14, 9]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [5, 14, 9]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0080/)

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





## 81. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 5, 7, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[6, 16, 12]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [6, 16, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0081/)

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





## 82. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 6, 8, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[7, 18, 6]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [7, 18, 6]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0082/)

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





## 83. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 7, 9, 2], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[8, 20, 9]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [8, 20, 9]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0083/)

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





## 84. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 8, 10, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[9, 22, 12]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [9, 22, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0084/)

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





## 85. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 9, 11, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[10, 24, 6]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [10, 24, 6]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0085/)

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





## 86. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 10, 12, 2], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[11, 26, 9]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [11, 26, 9]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0086/)

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





## 87. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 11, 13, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[12, 28, 12]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [12, 28, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0087/)

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





## 88. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 12, 14, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[13, 30, 6]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [13, 30, 6]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0088/)

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





## 89. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 13, 15, 2], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[14, 32, 9]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [14, 32, 9]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0089/)

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





## 90. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 14, 16, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[15, 34, 12]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [15, 34, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0090/)

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





## 91. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 15, 17, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[16, 36, 6]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [16, 36, 6]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0091/)

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





## 92. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 16, 18, 2], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[17, 38, 9]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [17, 38, 9]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0092/)

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





## 93. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 17, 19, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[18, 40, 12]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [18, 40, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0093/)

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





## 94. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 18, 20, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[19, 42, 6]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [19, 42, 6]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0094/)

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





## 95. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 19, 21, 2], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[20, 44, 9]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [20, 44, 9]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0095/)

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





## 96. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 20, 22, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[21, 46, 12]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [21, 46, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0096/)

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





## 97. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 21, 23, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[22, 48, 6]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [22, 48, 6]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0097/)

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





## 98. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 22, 24, 2], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[23, 50, 9]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [23, 50, 9]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0098/)

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





## 99. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 23, 25, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[24, 52, 12]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [24, 52, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0099/)

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





## 100. Complete Ascending Derivative Coefficients of a Sum

On all R, f has ascending coefficients [1, 24, 26, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.

View answer and explanation[25, 54, 6]

Add coefficientwise, differentiate, and remove trailing zero terms to obtain [25, 54, 6]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r04-derivative-rules-0100/)

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



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261007-advance-20261009-r04-derivative-rules-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261007-advance-20261009-r04-derivative-rules-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261007-advance-20261009-r04-derivative-rules-foundations/validation-report.json)