Derivative of \( \displaystyle - \frac{5 \cos{\left(4 x + 1 \right)}}{4} \)
Problem 2.986 · medium
Differentiate \( \displaystyle f(x) = - \frac{5 \cos{\left(4 x + 1 \right)}}{4} \).
- \[ \frac{d}{d x} \left(- \frac{5 \cos{\left(4 x + 1 \right)}}{4}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{5 \frac{d}{d x} \cos{\left(4 x + 1 \right)}}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{5 \sin{\left(4 x + 1 \right)} \frac{d}{d x} \left(4 x + 1\right)}{4} \]trigApply the derivative rule for cosine.✓ Proved
- \[ = 5 \sin{\left(4 x + 1 \right)} \]chain algebraApply the chain rule to the inner function. Simplify the expression by multiplying the constants.✓ Proved
Answer \( 5 \sin{\left(4 x + 1 \right)} \)
✓ 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 should be labeled 'chain'. Step 4 is labeled 'chain' but only differentiates the linear inner function, which is a basic 'derivative' step.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-27 — Step 3 is labeled 'trig' but applies the chain rule (differentiating the inner function), which should be labeled 'chain'. Step 4 is labeled 'chain' but only differentiates the linear inner function, which is a basic 'derivative' step.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (style) 2026-09-27 — Step 3 is labeled 'trig' but applies the chain rule (Derivative(cos(u), x) = -sin(u) * u'), which is a multi-rule step combining the trig derivative and the chain rule. Step 4 is then labeled 'chain' but only performs algebraic simplification of the inner derivative, making the label incorrect for the action performed.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.