Derivative of \( \displaystyle - \frac{\cos{\left(4 x + 2 \right)}}{4} \)
Problem 2.1002 · medium
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(4 x + 2 \right)}}{4} \).
- \[ \frac{d}{d x} \left(- \frac{\cos{\left(4 x + 2 \right)}}{4}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(4 x + 2 \right)}}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\sin{\left(4 x + 2 \right)} \frac{d}{d x} \left(4 x + 2\right)}{4} \]trigApply the derivative rule for cosine.✓ Proved
- \[ = \sin{\left(4 x + 2 \right)} \]chain simplifyApply the chain rule to the inner function. Simplify the resulting expression.✓ Proved
Answer \( \sin{\left(4 x + 2 \right)} \)
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 |
| 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 3 applies both the trig rule (derivative of cos) and the chain rule in a single step, violating the one‑rule‑per‑step rule. The label "trig" is also incomplete, as the inner derivative is handled by the chain rule.qwen3.6:27b-mlx: fail (style) — Step 3 applies both the trigonometric derivative rule and the chain rule simultaneously, violating the one-rule-per-step constraint. Step 4 is then redundant or mislabeled because the chain rule was already applied in Step 3.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-27 — Step 3 applies both the trigonometric derivative rule and the chain rule simultaneously, violating the one-rule-per-step constraint. Step 4 is then redundant or mislabeled because the chain rule was already applied in Step 3.gpt-oss:20b: fail (error) 2026-09-27 — Step 3 applies both the trig rule (derivative of cos) and the chain rule in a single step, violating the one‑rule‑per‑step rule. The label "trig" is also incomplete, as the inner derivative is handled by the chain rule.qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the constant multiple rule, the derivative of cosine, and the chain rule in separate steps. The labels are accurate and the logic is sound.gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.