Derivative of \( \displaystyle - \frac{\cos{\left(2 x + 2 \right)}}{2} \)
Problem 2.393 · medium
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(2 x + 2 \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\cos{\left(2 x + 2 \right)}}{2}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(2 x + 2 \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\sin{\left(2 x + 2 \right)} \frac{d}{d x} \left(2 x + 2\right)}{2} \]trigApply the derivative rule for cosine.✓ Proved
- \[ = \sin{\left(2 x + 2 \right)} \]derivative simplifyDifferentiate the inner function 2*x + 2. Simplify the resulting expression.✓ Proved
Answer \( \sin{\left(2 x + 2 \right)} \)
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 (multiplying by the derivative of the inner function) in a single step, violating the one‑rule‑per‑step rule. The label "trig" is also incomplete because the chain rule is used here.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (error) 2026-09-21 — Step 3 applies both the trig rule (derivative of cos) and the chain rule (multiplying by the derivative of the inner function) in a single step, violating the one‑rule‑per‑step rule. The label "trig" is also incomplete because the chain rule is used here.qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple rule, the chain rule (labeled as trig for the outer derivative), and basic derivative rules. Each step changes only one aspect of the expression, and the labels are appropriate.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple rule, chain rule (labeled as trig for the outer derivative), and derivative rule for the inner function, with correct algebraic simplification.gpt-oss:20b: fail (style) 2026-09-20 — Step 1 has no rule label, and step 3 incorrectly labels the chain rule as "trig". Both steps violate the requirement that each step must be labeled with a single, correct rule from the allowed vocabulary.
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.