Derivative of \( \displaystyle - \frac{\cos{\left(2 x + 2 \right)}}{2} \)
Problem 2.393 · medium Video
- \[ \frac{d}{d x} \left(- \frac{\cos{\left(2 x + 2 \right)}}{2}\right) \]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
✓ 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 |
| 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
Senior review claude-sonnet-5-5, 2026-10-08: pass — Every line is SymPy-verified equal to the previous one, the rule labels are legitimate, and the final result sin(2*x+2) matches the stated answer. Derivative of -cos(2x+2)/2 is indeed sin(2x+2).
gpt-oss:20b: dismiss — Differentiating cos(u) to -sin(u)*u' is a single chain-rule application via the trig derivative; the label 'trig' names a rule actually used, and the contract treats this as one step. Leaving Derivative(2*x+2, x) unevaluated shows the step does not do two independent things.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-08 — Every line is SymPy-verified equal to the previous one, the rule labels are legitimate, and the final result sin(2*x+2) matches the stated answer. Derivative of -cos(2x+2)/2 is indeed sin(2x+2).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.