Derivative of \( \displaystyle - \frac{\cos{\left(4 x + 1 \right)}}{2} \)
Problem 2.833 · medium
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(4 x + 1 \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\cos{\left(4 x + 1 \right)}}{2}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(4 x + 1 \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\sin{\left(4 x + 1 \right)} \frac{d}{d x} \left(4 x + 1\right)}{2} \]trigApply the derivative rule for cosine.✓ Proved
- \[ = 2 \sin{\left(4 x + 1 \right)} \]chain algebraApply the chain rule to the inner function. Simplify the coefficients.✓ Proved
Answer \( 2 \sin{\left(4 x + 1 \right)} \)
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: passqwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'trig' but actually applies the chain rule (differentiating the composite function cos(4x+1)). Step 4 is labeled 'chain' but only differentiates the inner linear function, which is a basic derivative step, not the chain rule itself. The labels for the chain rule application are swapped/misapplied.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 3 is labeled 'trig' but actually applies the chain rule (differentiating the composite function cos(4x+1)). Step 4 is labeled 'chain' but only differentiates the inner linear function, which is a basic derivative step, not the chain rule itself. The labels for the chain rule application are swapped/misapplied.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 3 is labeled 'trig' but actually applies the chain rule to differentiate the composite function cos(4x+1). Step 4 is labeled 'chain' but only differentiates the inner linear function, which is a basic derivative step, not the chain rule itself. The labels are swapped and incorrect.gpt-oss:20b: pass 2026-09-26
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.