Derivative of \( \displaystyle - \frac{\sin{\left(5 x \right)}}{5} \)
Problem 2.93 · medium Mental math
Differentiate \( \displaystyle f(x) = - \frac{\sin{\left(5 x \right)}}{5} \).
- \[ \frac{d}{d x} \left(- \frac{\sin{\left(5 x \right)}}{5}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \sin{\left(5 x \right)}}{5} \]constant-multiplePull out the constant factor -1/5.✓ Proved
- \[ = - \cos{\left(5 x \right)} \]derivative algebraApply the derivative rule for sine. Simplify the product of 1/5 and 5.✓ Proved
Answer \( - \cos{\left(5 x \right)} \)
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 |
| 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: passdeepseek-r1:70b: fail (misleading) — Step 3 should be labeled 'chain', not 'derivative', as it applies the chain rule by multiplying with the derivative of the inner function.qwen3.6:27b-mlx: fail (error) — Step 3 applies both the derivative of sine and the chain rule in a single step, violating the one-rule-per-step constraint. It should be split into applying the derivative of sine to get cos(5*x) and then applying the chain rule to multiply by the derivative of the inner function.
Every verdict on record (12)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 applies both the derivative of sine and the chain rule in a single step, violating the one-rule-per-step constraint. It should be split into applying the derivative of sine to get cos(5*x) and then applying the chain rule to multiply by the derivative of the inner function.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 is labeled 'derivative' but applies the chain rule to differentiate sin(5x). The label 'derivative' is reserved for basic derivatives (like d/dx sin(u) -> cos(u)), whereas handling the inner function 5x requires the 'chain' rule. Applying both the outer derivative and the inner derivative in a single step labeled only 'derivative' violates the one-rule-per-step constraint and mislabels the chain rule application.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 3 applies both the derivative of sine and the chain rule (differentiating the inner function 5x) in a single step, violating the one-rule-per-step constraint. It should be split into applying the derivative of sine to get cos(5x)*Derivative(5x, x) and then evaluating the inner derivative.deepseek-r1:70b: fail (misleading) 2026-09-19 — Step 3 should be labeled 'chain', not 'derivative', as it applies the chain rule by multiplying with the derivative of the inner function.gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: pass 2026-09-17deepseek-r1:70b: pass 2026-09-17
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.