Derivative of \( \displaystyle \left(5 x + 2\right) \sin{\left(5 x + 2 \right)} \)
Problem 2.517 · medium
Differentiate \( \displaystyle f(x) = \left(5 x + 2\right) \sin{\left(5 x + 2 \right)} \).
- \[ \frac{d}{d x} \left(5 x + 2\right) \sin{\left(5 x + 2 \right)} \]Start with the derivative of the function.✓ Proved
- \[ = \left(5 x + 2\right) \frac{d}{d x} \sin{\left(5 x + 2 \right)} + \sin{\left(5 x + 2 \right)} \frac{d}{d x} \left(5 x + 2\right) \]productApply the product rule.✓ Proved
- \[ = \left(5 x + 2\right) \frac{d}{d x} \sin{\left(5 x + 2 \right)} + 5 \sin{\left(5 x + 2 \right)} \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = \left(5 x + 2\right) \cos{\left(5 x + 2 \right)} \frac{d}{d x} \left(5 x + 2\right) + 5 \sin{\left(5 x + 2 \right)} \]trigDifferentiate the sine function.✓ Proved
- \[ = 5 \left(5 x + 2\right) \cos{\left(5 x + 2 \right)} + 5 \sin{\left(5 x + 2 \right)} \]derivativeDifferentiate the inner function 5*x + 2.✓ Proved
- \[ = \left(25 x + 10\right) \cos{\left(5 x + 2 \right)} + 5 \sin{\left(5 x + 2 \right)} \]algebraRearrange the terms.✓ Proved
Answer \( \left(25 x + 10\right) \cos{\left(5 x + 2 \right)} + 5 \sin{\left(5 x + 2 \right)} \)
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 4 is labeled 'trig' but applies the chain rule to differentiate sin(5x+2); the label should be 'chain'. Step 5 is labeled 'derivative' but computes the derivative of a linear term, which is better described as 'constant-multiple' or 'derivative' of the inner function, but the previous step already invoked the chain rule structure, making the separation awkward. More critically, Step 4 claims to 'Differentiate the sine function' but leaves the derivative of the inner function unevaluated, effectively splitting the chain rule application across steps 4 and 5 in a way that Step 4's label 'trig' is incorrect for the operation performed (which is applying the chain rule structure).gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (style) 2026-09-20 — Step 4 is labeled "trig" but it applies the chain rule to differentiate sin(5*x+2). The correct label should be "chain". No other steps combine multiple rules.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-09-26 with SymPy 1.14.0.