Derivative of \( \displaystyle \frac{\cos{\left(5 x \right)}}{5} \)
Problem 2.495 · easy
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(5 x \right)}}{5} \).
- \[ \frac{d}{d x} \frac{\cos{\left(5 x \right)}}{5} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \cos{\left(5 x \right)}}{5} \]constant-multiplePull out the constant factor 1/5.✓ Proved
- \[ = - \frac{\sin{\left(5 x \right)} \frac{d}{d x} 5 x}{5} \]chainApply the chain rule to the cosine function.✓ Proved
- \[ = - \sin{\left(5 x \right)} \]derivative simplifyDifferentiate the inner function 5*x. Simplify the expression by multiplying the constants.✓ Proved
Answer \( - \sin{\left(5 x \right)} \)
✓ 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: fail (style) — Step 1 lacks a rule label; it should be marked "derivative". No other defects are present.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (style) 2026-09-21 — Step 1 lacks a rule label; it should be marked "derivative". No other defects are present.qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple rule, chain rule, and basic derivative rules in separate steps. The labels are accurate and the logic is sound.gpt-oss:20b: fail (style) 2026-09-20 — Step 1 is missing a rule label; it should be labeled "derivative" to indicate the application of the derivative rule.
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.