Derivative of \( \displaystyle - \frac{x}{2} + \frac{\sin{\left(4 x \right)}}{8} \)
Problem 2.606 · hard Video
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(4 x \right)}}{8} \).
- \[ \frac{d}{d x} \left(- \frac{x}{2} + \frac{\sin{\left(4 x \right)}}{8}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \frac{\sin{\left(4 x \right)}}{8} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{\frac{d}{d x} \sin{\left(4 x \right)}}{8} \]constantFactor out the constant 1/8.✓ Proved
- \[ = - \frac{d}{d x} \frac{x}{2} + \frac{\frac{d}{d x} \sin{\left(4 x \right)}}{8} \]constantFactor out the constant -1/2.✓ Proved
- \[ = - \frac{\frac{d}{d x} x}{2} + \frac{\frac{d}{d x} \sin{\left(4 x \right)}}{8} \]constant-multipleSeparate the constant from the variable.✓ Proved
- \[ = \frac{\frac{d}{d x} \sin{\left(4 x \right)}}{8} - \frac{1}{2} \]derivative algebraDifferentiate x. Simplify the first term.✓ Proved
- \[ = \frac{\cos{\left(4 x \right)}}{2} - \frac{1}{2} \]trig algebraApply the chain rule to sin(4*x). Simplify the fraction 4/8.✓ Proved
Answer \( \frac{\cos{\left(4 x \right)} - 1}{2} \)
✓ 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. How we verify
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: fail (error) — Step 8 applies two rules at once (derivative of sin and the chain rule) and labels the operation as "trig" instead of the correct rule name. This violates the one‑rule‑per‑step requirement and mislabels the applied rule.qwen3.6:27b-mlx: fail (style) — Step 8 applies the chain rule to differentiate sin(4*x), but is labeled 'trig'. The label 'trig' is reserved for trigonometric identities, not differentiation rules; 'chain' is the correct label from the vocabulary.
Senior review claude-sonnet-5-5, 2026-10-10: pass — Every line is a correct equivalence, SymPy proved each one, and the final line matches the stated answer. The labels name rules that were actually applied, including 'trig' on step 8.
gpt-oss:20b: dismiss — Differentiating sin(4x) to cos(4x)*4 is one chain-rule step, since evaluating the simple inner derivative is part of that step, and 'trig' legitimately names the sin-to-cos derivative that was applied.qwen3.6:27b-mlx: dismiss — The step applies the derivative of sine, so 'trig' names a rule actually used; 'chain' would also fit, and the contract accepts either label when two fit.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-10 — Every line is a correct equivalence, SymPy proved each one, and the final line matches the stated answer. The labels name rules that were actually applied, including 'trig' on step 8.qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 8 applies the chain rule to differentiate sin(4*x), but is labeled 'trig'. The label 'trig' is reserved for trigonometric identities, not differentiation rules; 'chain' is the correct label from the vocabulary.gpt-oss:20b: fail (error) 2026-09-26 — Step 8 applies two rules at once (derivative of sin and the chain rule) and labels the operation as "trig" instead of the correct rule name. This violates the one‑rule‑per‑step requirement and mislabels the applied rule.qwen3.6:27b-mlx: fail (misleading) 2026-09-21 — Step 8 applies the chain rule but is labeled 'trig'. The label 'trig' is reserved for trigonometric identities or basic derivatives, whereas this step explicitly requires the chain rule for the inner function 4x. Using 'trig' obscures the application of the chain rule, which is a distinct and necessary rule here.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 8 applies the chain rule but is labeled 'trig'; the label 'chain' is available in the vocabulary and more accurately describes the operation. Step 4 factors out a negative sign, which is better labeled 'algebra' or 'simplify' rather than 'constant'.gpt-oss:20b: fail (style) 2026-09-21 — Step 8 incorrectly labels the application of the chain rule as "trig"; the correct rule name is "chain".
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.