Derivative of \( \displaystyle \frac{5 x}{2} - \frac{5 \sin{\left(6 x - 6 \right)}}{12} \)
Problem 2.1356 · hard
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(6 x - 6 \right)}}{12} \).
- \[ \frac{d}{d x} \left(\frac{5 x}{2} - \frac{5 \sin{\left(6 x - 6 \right)}}{12}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{d}{d x} \frac{5 \sin{\left(6 x - 6 \right)}}{12} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{5 \frac{d}{d x} \sin{\left(6 x - 6 \right)}}{12} \]constant-multipleFactor out the constant from the second term.✓ Proved
- \[ = - \frac{5 \cos{\left(6 x - 6 \right)} \frac{d}{d x} \left(6 x - 6\right)}{12} + \frac{d}{d x} \frac{5 x}{2} \]trigApply the derivative of the sine function.✓ Proved
- \[ = - \frac{5 \cos{\left(6 x - 6 \right)} \frac{d}{d x} 6 x}{12} + \frac{d}{d x} \frac{5 x}{2} \]sumApply the chain rule to the inner function.✓ Proved
- \[ = - \frac{5 \cos{\left(6 x - 6 \right)} \frac{d}{d x} x}{2} + \frac{d}{d x} \frac{5 x}{2} \]constant-multiple algebraFactor out the constant 6. Simplify the coefficient (5/12)*6.✓ Proved
- \[ = - \frac{5 \cos{\left(6 x - 6 \right)}}{2} + \frac{d}{d x} \frac{5 x}{2} \]derivativeDifferentiate x.✓ Proved
- \[ = \frac{5}{2} - \frac{5 \cos{\left(6 x - 6 \right)}}{2} \]simplifySimplify the final expression.✓ Proved
Answer \( \frac{5}{2} - \frac{5 \cos{\left(6 x - 6 \right)}}{2} \)
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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ 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 5 is labeled 'sum' but applies the chain rule to the inner function of the sine term. Additionally, Step 4 is labeled 'trig' but effectively applies the chain rule (differentiating the outer function and multiplying by the derivative of the inner), which should be labeled 'chain'.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 5 is labeled 'sum' but applies the chain rule to the inner function of the sine term. Additionally, Step 4 is labeled 'trig' but effectively applies the chain rule (differentiating the outer function and multiplying by the derivative of the inner), which should be labeled 'chain'.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-09-30 — Step 5 is labeled 'sum' but applies the chain rule to the inner function 6*x - 6. The label does not match the rule applied.gpt-oss:20b: fail (error) 2026-09-30 — Step 5 incorrectly drops the derivative of the constant term – it replaces Derivative(6*x - 6, x) with Derivative(6*x, x), omitting the zero contribution from the –6. This makes the chain‑rule application mathematically incorrect.
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 2026-10-03 with SymPy 1.14.0.