Derivative of \( \displaystyle - \frac{\cos{\left(5 x - 3 \right)}}{5} \)
Problem 2.1351 · medium
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(5 x - 3 \right)}}{5} \).
- \[ \frac{d}{d x} \left(- \frac{\cos{\left(5 x - 3 \right)}}{5}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(5 x - 3 \right)}}{5} \]constantPull out the constant factor.✓ Proved
- \[ = \frac{\sin{\left(5 x - 3 \right)} \frac{d}{d x} \left(5 x - 3\right)}{5} \]trigApply the derivative rule for cosine.✓ Proved
- \[ = \sin{\left(5 x - 3 \right)} \]chain simplifyApply the chain rule to the inner function. Simplify the resulting expression.✓ Proved
Answer \( \sin{\left(5 x - 3 \right)} \)
✓ 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 |
| 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: fail (style) — Step 3 is labeled 'trig' but applies the chain rule (differentiating the inner function), which is a mislabeling. Step 4 is labeled 'chain' but only performs algebraic simplification of the derivative of the linear term, which is a mislabeling.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-03 — Step 3 is labeled 'trig' but applies the chain rule (differentiating the inner function), which is a mislabeling. Step 4 is labeled 'chain' but only performs algebraic simplification of the derivative of the linear term, which is a mislabeling.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (style) 2026-09-30 — Step 4 is labeled 'chain', but the chain rule was already applied in Step 3 (which produced the derivative of the inner function). Step 4 merely computes the derivative of the linear inner function (5*x - 3), which should be labeled 'derivative' or 'algebra', not 'chain'. Additionally, Step 3 applies the chain rule structure but labels it 'trig'; while 'trig' is acceptable for the outer derivative, the step effectively combines the trig derivative and the chain rule setup, which is borderline, but the main defect is the mislabeling of Step 4.gpt-oss:20b: pass 2026-09-30
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 2026-10-03 with SymPy 1.14.0.