Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4} \)
Problem 2.34 · hard Beautiful Video
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4} \).
- \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(2 x \right)}}{4} \]sumApply the sum rule for derivatives.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(2 x \right)}}{4} \]constantPull out the constant factor from the second term.✓ Proved
- \[ = - \frac{\cos{\left(2 x \right)} \frac{d}{d x} 2 x}{4} + \frac{d}{d x} \frac{x}{2} \]chainApply the chain rule to the sine term.✓ Proved
- \[ = \frac{1}{2} - \frac{\cos{\left(2 x \right)}}{2} \]derivative algebra simplifyEvaluate the derivatives of the individual terms. Multiply the constants. Simplify the fraction.✓ Proved
Answer \( \sin^{2}{\left(x \right)} \)
✓ 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. 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 |
| 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) — The solution stops at 1/2 - cos(2x)/2 but should apply the double-angle identity to reach sin(x)^2, which is the stated answer.qwen3.6:27b-mlx: fail (error) — Step 5 applies the derivative rule to two distinct terms simultaneously (x/2 and 2*x), violating the one-change-per-step constraint. Additionally, the stated answer sin(x)**2 is not algebraically equal to the final line 1/2 - cos(2*x)/2 without a further simplification step using a trigonometric identity, which is missing.
Senior review claude-sonnet-5-5, 2026-10-03: pass — Every step is correct and checked, and the final line is verified equal to sin(x)^2. The solution stops at 1/2 - cos(2x)/2 without an explicit trig step to sin(x)^2, and step 3's 'constant' label would read more precisely as 'constant-multiple'. Both are minor style points, not defects.
qwen3.6:27b-mlx: dismiss — Step 5 applies the single derivative rule (d/dx of x/2 and of 2x) to simple terms, which counts as one change. The final line 1/2 - cos(2x)/2 is proved equal to the stated answer sin(x)^2 by SymPy, so the unstated half-angle identity is not a false statement or a wrong label.agent:pipeline: uphold — Beautiful nomination: The messy x/2 - sin(2x)/4 differentiates to 1/2 - cos(2x)/2, which collapses by the half-angle identity to sin(x)^2.
Every verdict on record (13)
claude-sonnet-5-5: pass 2026-10-03 — Every step is correct and checked, and the final line is verified equal to sin(x)^2. The solution stops at 1/2 - cos(2x)/2 without an explicit trig step to sin(x)^2, and step 3's 'constant' label would read more precisely as 'constant-multiple'. Both are minor style points, not defects.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 5 applies the derivative rule to two distinct terms simultaneously (x/2 and 2*x), violating the one-change-per-step constraint. Additionally, the stated answer sin(x)**2 is not algebraically equal to the final line 1/2 - cos(2*x)/2 without a further simplification step using a trigonometric identity, which is missing.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 5 applies two rules at once: it evaluates the derivative of x/2 (derivative/constant-multiple) and the derivative of 2*x (derivative) simultaneously, violating the one-rule-per-step constraint.gpt-oss:20b: fail (error) 2026-09-20 — The final answer given (sin(x)**2) does not match the correct derivative (1/2 - cos(2*x)/2). The solution steps are mathematically correct, but the stated answer is incorrect.qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: fail (misleading) 2026-09-19 — The solution stops at 1/2 - cos(2x)/2 but should apply the double-angle identity to reach sin(x)^2, which is the stated answer.gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The final result is algebraically equivalent to the stated answer.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — The final answer sin(x)**2 does not equal the computed derivative 1/2 - cos(2*x)/2. The solution incorrectly claims the derivative is sin(x)**2, which is mathematically wrong.gpt-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.