Polynomial Differentiation Rule Foundations
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batch-20261007-advance-20261008-r05-derivative-rules-foundations · 2026-10-07 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [3, 1], starting with its constant term; g has ascending coefficients [-5, -1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
0
Use f′+g′. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
2. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [-11, 2, 0, 2], starting with its constant term; g has ascending coefficients [1, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
8
Use f′+g′. The exact result is 8; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
3. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [-10, 3, 1, 3], starting with its constant term; g has ascending coefficients [0, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
2
Use f′+g′. The exact result is 2; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
4. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term; g has ascending coefficients [-1, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
18
Use f′+g′. The exact result is 18; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
5. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term; g has ascending coefficients [-2, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
8
Use f′+g′. The exact result is 8; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
6. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [-7, 6, 1, 2], starting with its constant term; g has ascending coefficients [-3, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
33
Use f′+g′. The exact result is 33; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
7. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [-6, 7, -1, 3], starting with its constant term; g has ascending coefficients [-4, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
12
Use f′+g′. The exact result is 12; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
8. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [-5, 8, 0, 4], starting with its constant term; g has ascending coefficients [-5, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
7
Use f′+g′. The exact result is 7; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
9. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term; g has ascending coefficients [-6, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
12
Use f′+g′. The exact result is 12; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
10. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term; g has ascending coefficients [-7, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
45
Use f′+g′. The exact result is 45; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
11. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term; g has ascending coefficients [-8, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
30
Use f′+g′. The exact result is 30; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
12. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [-1, 12, 1, 4], starting with its constant term; g has ascending coefficients [-9, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
22
Use f′+g′. The exact result is 22; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
13. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term; g has ascending coefficients [-10, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
12
Use f′+g′. The exact result is 12; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
14. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term; g has ascending coefficients [-11, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
24
Use f′+g′. The exact result is 24; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
15. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [2, 15, 1, 3], starting with its constant term; g has ascending coefficients [-12, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
46
Use f′+g′. The exact result is 46; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
16. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [3, 16, -1, 4], starting with its constant term; g has ascending coefficients [-13, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
75
Use f′+g′. The exact result is 75; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
17. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [4, 17, 0, 1], starting with its constant term; g has ascending coefficients [-14, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
14
Use f′+g′. The exact result is 14; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
18. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term; g has ascending coefficients [-15, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
17
Use f′+g′. The exact result is 17; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
19. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term; g has ascending coefficients [-16, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
24
Use f′+g′. The exact result is 24; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
20. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [7, 20, 0, 4], starting with its constant term; g has ascending coefficients [-17, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
83
Use f′+g′. The exact result is 83; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
21. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term; g has ascending coefficients [-18, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
12
Use f′+g′. The exact result is 12; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
22. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [9, 22, -1, 2], starting with its constant term; g has ascending coefficients [-19, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
30
Use f′+g′. The exact result is 30; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
23. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [10, 23, 0, 3], starting with its constant term; g has ascending coefficients [-20, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
22
Use f′+g′. The exact result is 22; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
24. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term; g has ascending coefficients [-21, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
42
Use f′+g′. The exact result is 42; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
25. Derivative of a Polynomial Sum
On all R, f has ascending coefficients [12, 25, -1, 1], starting with its constant term; g has ascending coefficients [-22, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
24
Use f′+g′. The exact result is 24; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-sum · candidate
26. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [1, 1], starting with its constant term. Find the derivative of (0)f at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
0
Use c*f′, including c=0. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
27. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [-11, 2, 0, 2], starting with its constant term. Find the derivative of (-2)f at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-16
Use c*f′, including c=0. The exact result is -16; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
28. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [-10, 3, 1, 3], starting with its constant term. Find the derivative of (3)f at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
9
Use c*f′, including c=0. The exact result is 9; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
29. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term. Find the derivative of (-4)f at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-56
Use c*f′, including c=0. The exact result is -56; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
30. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term. Find the derivative of (1)f at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
17
Use c*f′, including c=0. The exact result is 17; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
31. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [-7, 6, 1, 2], starting with its constant term. Find the derivative of (-2)f at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-52
Use c*f′, including c=0. The exact result is -52; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
32. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [-6, 7, -1, 3], starting with its constant term. Find the derivative of (3)f at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
54
Use c*f′, including c=0. The exact result is 54; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
33. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [-5, 8, 0, 4], starting with its constant term. Find the derivative of (-4)f at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-32
Use c*f′, including c=0. The exact result is -32; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
34. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term. Find the derivative of (1)f at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
14
Use c*f′, including c=0. The exact result is 14; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
35. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term. Find the derivative of (-2)f at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-60
Use c*f′, including c=0. The exact result is -60; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
36. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term. Find the derivative of (3)f at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
141
Use c*f′, including c=0. The exact result is 141; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
37. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [-1, 12, 1, 4], starting with its constant term. Find the derivative of (-4)f at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-88
Use c*f′, including c=0. The exact result is -88; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
38. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term. Find the derivative of (1)f at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
13
Use c*f′, including c=0. The exact result is 13; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
39. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term. Find the derivative of (-2)f at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-40
Use c*f′, including c=0. The exact result is -40; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
40. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [2, 15, 1, 3], starting with its constant term. Find the derivative of (3)f at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
165
Use c*f′, including c=0. The exact result is 165; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
41. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [3, 16, -1, 4], starting with its constant term. Find the derivative of (-4)f at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-272
Use c*f′, including c=0. The exact result is -272; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
42. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [4, 17, 0, 1], starting with its constant term. Find the derivative of (1)f at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
20
Use c*f′, including c=0. The exact result is 20; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
43. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term. Find the derivative of (-2)f at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-36
Use c*f′, including c=0. The exact result is -36; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
44. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term. Find the derivative of (3)f at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
78
Use c*f′, including c=0. The exact result is 78; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
45. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [7, 20, 0, 4], starting with its constant term. Find the derivative of (-4)f at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-272
Use c*f′, including c=0. The exact result is -272; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
46. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term. Find the derivative of (1)f at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
29
Use c*f′, including c=0. The exact result is 29; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
47. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [9, 22, -1, 2], starting with its constant term. Find the derivative of (-2)f at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-60
Use c*f′, including c=0. The exact result is -60; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
48. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [10, 23, 0, 3], starting with its constant term. Find the derivative of (3)f at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
69
Use c*f′, including c=0. The exact result is 69; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
49. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term. Find the derivative of (-4)f at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-152
Use c*f′, including c=0. The exact result is -152; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
50. Derivative of a Scalar Multiple of a Polynomial
On all R, f has ascending coefficients [12, 25, -1, 1], starting with its constant term. Find the derivative of (1)f at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
33
Use c*f′, including c=0. The exact result is 33; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-scalar · candidate
51. Derivative of a Polynomial Product
On all R, f has ascending coefficients [3], starting with its constant term; g has ascending coefficients [-2], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
0
Use f′g+fg′, not f′g′. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
52. Derivative of a Polynomial Product
On all R, f has ascending coefficients [-11, 2, 0, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 3], starting with its constant term. Find the derivative of fg at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-105
Use f′g+fg′, not f′g′. The exact result is -105; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
53. Derivative of a Polynomial Product
On all R, f has ascending coefficients [-10, 3, 1, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 4], starting with its constant term. Find the derivative of fg at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-4
Use f′g+fg′, not f′g′. The exact result is -4; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
54. Derivative of a Polynomial Product
On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 2], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
78
Use f′g+fg′, not f′g′. The exact result is 78; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
55. Derivative of a Polynomial Product
On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 3], starting with its constant term. Find the derivative of fg at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
866
Use f′g+fg′, not f′g′. The exact result is 866; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
56. Derivative of a Polynomial Product
On all R, f has ascending coefficients [-7, 6, 1, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 4], starting with its constant term. Find the derivative of fg at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-2040
Use f′g+fg′, not f′g′. The exact result is -2040; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
57. Derivative of a Polynomial Product
On all R, f has ascending coefficients [-6, 7, -1, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 2], starting with its constant term. Find the derivative of fg at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-85
Use f′g+fg′, not f′g′. The exact result is -85; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
58. Derivative of a Polynomial Product
On all R, f has ascending coefficients [-5, 8, 0, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 3], starting with its constant term. Find the derivative of fg at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
6
Use f′g+fg′, not f′g′. The exact result is 6; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
59. Derivative of a Polynomial Product
On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 4], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
175
Use f′g+fg′, not f′g′. The exact result is 175; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
60. Derivative of a Polynomial Product
On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 2], starting with its constant term. Find the derivative of fg at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
1472
Use f′g+fg′, not f′g′. The exact result is 1472; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
61. Derivative of a Polynomial Product
On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 3], starting with its constant term. Find the derivative of fg at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-2524
Use f′g+fg′, not f′g′. The exact result is -2524; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
62. Derivative of a Polynomial Product
On all R, f has ascending coefficients [-1, 12, 1, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 4], starting with its constant term. Find the derivative of fg at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-258
Use f′g+fg′, not f′g′. The exact result is -258; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
63. Derivative of a Polynomial Product
On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 2], starting with its constant term. Find the derivative of fg at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
26
Use f′g+fg′, not f′g′. The exact result is 26; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
64. Derivative of a Polynomial Product
On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 3], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
307
Use f′g+fg′, not f′g′. The exact result is 307; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
65. Derivative of a Polynomial Product
On all R, f has ascending coefficients [2, 15, 1, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 4], starting with its constant term. Find the derivative of fg at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
5380
Use f′g+fg′, not f′g′. The exact result is 5380; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
66. Derivative of a Polynomial Product
On all R, f has ascending coefficients [3, 16, -1, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 2], starting with its constant term. Find the derivative of fg at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-2382
Use f′g+fg′, not f′g′. The exact result is -2382; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
67. Derivative of a Polynomial Product
On all R, f has ascending coefficients [4, 17, 0, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 3], starting with its constant term. Find the derivative of fg at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-64
Use f′g+fg′, not f′g′. The exact result is -64; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
68. Derivative of a Polynomial Product
On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 4], starting with its constant term. Find the derivative of fg at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
36
Use f′g+fg′, not f′g′. The exact result is 36; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
69. Derivative of a Polynomial Product
On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 2], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
399
Use f′g+fg′, not f′g′. The exact result is 399; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
70. Derivative of a Polynomial Product
On all R, f has ascending coefficients [7, 20, 0, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 3], starting with its constant term. Find the derivative of fg at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
5630
Use f′g+fg′, not f′g′. The exact result is 5630; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
71. Derivative of a Polynomial Product
On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 4], starting with its constant term. Find the derivative of fg at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-2330
Use f′g+fg′, not f′g′. The exact result is -2330; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
72. Derivative of a Polynomial Product
On all R, f has ascending coefficients [9, 22, -1, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 2], starting with its constant term. Find the derivative of fg at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-34
Use f′g+fg′, not f′g′. The exact result is -34; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
73. Derivative of a Polynomial Product
On all R, f has ascending coefficients [10, 23, 0, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 3], starting with its constant term. Find the derivative of fg at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
56
Use f′g+fg′, not f′g′. The exact result is 56; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
74. Derivative of a Polynomial Product
On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 4], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
982
Use f′g+fg′, not f′g′. The exact result is 982; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
75. Derivative of a Polynomial Product
On all R, f has ascending coefficients [12, 25, -1, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 2], starting with its constant term. Find the derivative of fg at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
2442
Use f′g+fg′, not f′g′. The exact result is 2442; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-product · candidate
76. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [1, 2, 3], starting with its constant term; g has ascending coefficients [4], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
0
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
77. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [-11, 2, 1, 1], starting with its constant term; g has ascending coefficients [-4, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
42
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 42; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
78. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [-10, 3, 2, 1], starting with its constant term; g has ascending coefficients [-3, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
230
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 230; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
79. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [-9, 4, 0, 1], starting with its constant term; g has ascending coefficients [-2, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
7412
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 7412; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
80. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [-8, 1, 1, 1], starting with its constant term; g has ascending coefficients [-1, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
0
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
81. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [-7, 2, 2, 1], starting with its constant term; g has ascending coefficients [0, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
2
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 2; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
82. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [-6, 3, 0, 1], starting with its constant term; g has ascending coefficients [1, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
306
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 306; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
83. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [-5, 4, 1, 1], starting with its constant term; g has ascending coefficients [2, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
26124
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 26124; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
84. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [-4, 1, 2, 1], starting with its constant term; g has ascending coefficients [3, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-130
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is -130; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
85. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [-3, 2, 0, 1], starting with its constant term; g has ascending coefficients [4, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
50
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 50; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
86. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [-2, 3, 1, 1], starting with its constant term; g has ascending coefficients [5, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
2112
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 2112; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
87. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [-1, 4, 2, 1], starting with its constant term; g has ascending coefficients [6, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
61700
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 61700; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
88. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [0, 1, 0, 1], starting with its constant term; g has ascending coefficients [7, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
218
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 218; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
89. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [1, 2, 1, 1], starting with its constant term; g has ascending coefficients [8, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
210
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 210; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
90. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [2, 3, 2, 1], starting with its constant term; g has ascending coefficients [9, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
6470
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 6470; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
91. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [3, 4, 0, 1], starting with its constant term; g has ascending coefficients [10, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
29444
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 29444; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
92. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [4, 1, 1, 1], starting with its constant term; g has ascending coefficients [11, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
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0
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
93. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [5, 2, 2, 1], starting with its constant term; g has ascending coefficients [12, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
482
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 482; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
94. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [6, 3, 0, 1], starting with its constant term; g has ascending coefficients [13, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
4626
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 4626; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
95. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [7, 4, 1, 1], starting with its constant term; g has ascending coefficients [14, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
65940
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 65940; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
96. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [8, 1, 2, 1], starting with its constant term; g has ascending coefficients [15, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
-1666
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is -1666; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
97. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [9, 2, 0, 1], starting with its constant term; g has ascending coefficients [16, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
770
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 770; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
98. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [10, 3, 1, 1], starting with its constant term; g has ascending coefficients [17, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
10944
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 10944; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
99. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [11, 4, 2, 1], starting with its constant term; g has ascending coefficients [18, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
124100
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 124100; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
100. Chain-Rule Derivative of a Polynomial Composition
On all R, f has ascending coefficients [12, 1, 0, 1], starting with its constant term; g has ascending coefficients [19, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.
View answer and explanation
1946
Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 1946; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions.
Reviewed step-by-step derivation
Differentiating polynomial sums, products, and compositions · qf-derivative-rule-chain · candidate
Canonical question JSONL · English JSONL · Validation report