Derivative of \( \displaystyle 9 x^{2} \cos{\left(3 x \right)} \)
Problem 2.575 · hard Video
Differentiate \( \displaystyle f(x) = 9 x^{2} \cos{\left(3 x \right)} \).
- \[ \frac{d}{d x} 9 x^{2} \cos{\left(3 x \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 9 \frac{d}{d x} x^{2} \cos{\left(3 x \right)} \]constant-multiplePull out the constant factor 9.✓ Proved
- \[ = 9 x^{2} \frac{d}{d x} \cos{\left(3 x \right)} + 9 \cos{\left(3 x \right)} \frac{d}{d x} x^{2} \]productApply the product rule.✓ Proved
- \[ = 9 x^{2} \frac{d}{d x} \cos{\left(3 x \right)} + 18 x \cos{\left(3 x \right)} \]powerDifferentiate x**2.✓ Proved
- \[ = - 9 x^{2} \sin{\left(3 x \right)} \frac{d}{d x} 3 x + 18 x \cos{\left(3 x \right)} \]trigDifferentiate cos(3*x) using the chain rule.✓ Proved
- \[ = - 27 x^{2} \sin{\left(3 x \right)} + 18 x \cos{\left(3 x \right)} \]derivative algebra simplifyDifferentiate 3*x. Simplify the expression inside the parentheses. Distribute the 9 to get the final answer.✓ Proved
Answer \( - 27 x^{2} \sin{\left(3 x \right)} + 18 x \cos{\left(3 x \right)} \)
✓ 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 (style) — Step 5 applies the chain rule to differentiate cos(3*x) but labels it as "trig". The correct label for that step is "chain" (or "product" if you view the inner derivative as a product), not "trig".qwen3.6:27b-mlx: pass
Senior review claude-sonnet-5-5, 2026-10-10: pass — Every line is correct and SymPy proved each step. Each step changes one thing and its label names a rule it legitimately applied. The final line matches the stated answer.
gpt-oss:20b: dismiss — Step 5 differentiates cos(3x) to -sin(3x)*d(3x)/dx, which legitimately applies the trig rule (and the chain rule); when two labels fit, either is acceptable.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-10 — Every line is correct and SymPy proved each step. Each step changes one thing and its label names a rule it legitimately applied. The final line matches the stated answer.qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: fail (style) 2026-09-26 — Step 5 applies the chain rule to differentiate cos(3*x) but labels it as "trig". The correct label for that step is "chain" (or "product" if you view the inner derivative as a product), not "trig".qwen3.6:27b-mlx: fail (misleading) 2026-09-21 — Step 5 applies the chain rule to differentiate cos(3*x), but labels the step 'trig'. The label 'trig' typically refers to trigonometric identities or simplifications, not the chain rule. This is misleading because the core operation performed is the chain rule, not a trigonometric identity.gpt-oss:20b: fail (style) 2026-09-21 — Step 5 incorrectly labels the application of the chain rule as "trig". The derivative of cos(3*x) should be identified with the chain rule, not the trig rule, and the label should be "chain".qwen3.6:27b-mlx: fail (misleading) 2026-09-21 — Step 5 applies the chain rule to differentiate cos(3*x), but is labeled 'trig'. The label 'trig' typically refers to trigonometric identities or basic derivatives, whereas the step explicitly invokes the inner derivative, making 'chain' the correct label from the vocabulary.gpt-oss:20b: pass 2026-09-21
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.