Derivative of \( \displaystyle 5 x \sin{\left(5 x \right)} \)
Problem 2.584 · medium Video
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 \( 25 x \cos{\left(5 x \right)} + 5 \sin{\left(5 x \right)} \)
✓ 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. How we verify
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 (error) — Step 5 applies two rules at once: it uses the chain rule to differentiate sin(5*x) and the trig rule to differentiate the sine itself. The step should be split into separate chain and trig applications, or at least labeled with a rule that covers both, but the current single label "trig" is insufficient.qwen3.6:27b-mlx: fail (misleading) — Step 5 applies both the derivative of sine and the chain rule simultaneously, violating the one-rule-per-step constraint. Additionally, the label 'trig' is incorrect for a differentiation step; it should be 'chain' or 'derivative'.
Senior review claude-sonnet-5-5, 2026-10-10: pass — The derivation is correct. Every line was proved equivalent by SymPy, and the final line matches the stated answer. The labels are acceptable, and step 5 is a single chain-rule/trig step on one term.
gpt-oss:20b: dismiss — Differentiating sin(5*x) as cos(5*x)*d(5x) is a single chain-rule step on one term, and the contract counts evaluating the sine derivative as part of it; 'trig' names a rule the step legitimately applied.qwen3.6:27b-mlx: dismiss — The step changes only one thing (the derivative of sin(5*x)), and 'trig' is a legitimate label for the sine derivative; when both 'chain' and 'trig' fit, either is acceptable.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-10 — The derivation is correct. Every line was proved equivalent by SymPy, and the final line matches the stated answer. The labels are acceptable, and step 5 is a single chain-rule/trig step on one term.qwen3.6:27b-mlx: fail (misleading) 2026-09-26 — Step 5 applies both the derivative of sine and the chain rule simultaneously, violating the one-rule-per-step constraint. Additionally, the label 'trig' is incorrect for a differentiation step; it should be 'chain' or 'derivative'.gpt-oss:20b: fail (error) 2026-09-26 — Step 5 applies two rules at once: it uses the chain rule to differentiate sin(5*x) and the trig rule to differentiate the sine itself. The step should be split into separate chain and trig applications, or at least labeled with a rule that covers both, but the current single label "trig" is insufficient.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.