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 explanation
0/1
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
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 explanation
4/1
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
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 explanation
6/1
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
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 explanation
2/1
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
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 explanation
10/1
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
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 explanation
0/1
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
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 explanation
20/1
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
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 explanation
16/1
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
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 explanation
24/1
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
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 explanation
8/1
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
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 explanation
22/1
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
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 explanation
18/1
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
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 explanation
26/1
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
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 explanation
28/1
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
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 explanation
42/1
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
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 explanation
44/1
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
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 explanation
34/1
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
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 explanation
36/1
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
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 explanation
32/1
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
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 explanation
40/1
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
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 explanation
30/1
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
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 explanation
50/1
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
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 explanation
46/1
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
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 explanation
54/1
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
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 explanation
38/1
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
Yes
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Differentiating polynomial sums, products, and compositions · qf-density-derivative-sum-coefficients · candidate
Canonical question JSONL · English JSONL · Validation report