Derivative of \( \displaystyle 5 x \sin{\left(5 x \right)} \)
Problem 2.584 · medium
Differentiate \( \displaystyle f(x) = 5 x \sin{\left(5 x \right)} \).
- \[ \frac{d}{d x} 5 x \sin{\left(5 x \right)} \]Start with the derivative of the function.✓ Proved
- \[ = 5 \frac{d}{d x} x \sin{\left(5 x \right)} \]constant-multiplePull out the constant factor 5.✓ Proved
- \[ = 5 x \frac{d}{d x} \sin{\left(5 x \right)} + 5 \sin{\left(5 x \right)} \frac{d}{d x} x \]productApply the product rule to x*sin(5*x).✓ Proved
- \[ = 5 x \frac{d}{d x} \sin{\left(5 x \right)} + 5 \sin{\left(5 x \right)} \]derivativeDifferentiate x.✓ Proved
- \[ = 5 x \cos{\left(5 x \right)} \frac{d}{d x} 5 x + 5 \sin{\left(5 x \right)} \]trigDifferentiate sin(5*x) using the chain rule.✓ Proved
- \[ = 25 x \cos{\left(5 x \right)} + 5 \sin{\left(5 x \right)} \]derivative algebra simplifyDifferentiate 5*x. Rearrange the terms. Distribute the 5 to get the final answer.✓ Proved
Answer \( 25 x \cos{\left(5 x \right)} + 5 \sin{\left(5 x \right)} \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ 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: fail (style) — Step 5 is labeled "trig" but it applies the chain rule to differentiate sin(5*x). The correct label should be "chain" (or "derivative" if the vocabulary allows).qwen3.6:27b-mlx: fail (style) — Step 5 applies both the trigonometric derivative rule and the chain rule simultaneously, violating the one-rule-per-step constraint. It should be split into a step applying the trig rule to get cos(5*x)*Derivative(5*x, x) and a subsequent step applying the chain rule.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 5 applies both the trigonometric derivative rule and the chain rule simultaneously, violating the one-rule-per-step constraint. It should be split into a step applying the trig rule to get cos(5*x)*Derivative(5*x, x) and a subsequent step applying the chain rule.gpt-oss:20b: fail (style) 2026-09-21 — Step 5 is labeled "trig" but it applies the chain rule to differentiate sin(5*x). The correct label should be "chain" (or "derivative" if the vocabulary allows).qwen3.6:27b-mlx: fail (misleading) 2026-09-21 — Step 5 applies the chain rule but is labeled 'trig'. The label 'trig' is not in the allowed vocabulary, and even if it were, the step fundamentally relies on the chain rule, not just a trigonometric identity.gpt-oss:20b: fail (style) 2026-09-21 — Step 5 incorrectly labels the application of the chain rule as "trig"; the correct label from the allowed vocabulary is "chain". No other steps violate the one‑rule‑per‑step rule.
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.