Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(10 x \right)}}{20} \)
Problem 2.856 · hard Video
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(10 x \right)}}{20} \).
- \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(10 x \right)}}{20}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(10 x \right)}}{20} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(10 x \right)}}{20} \]constant-multipleFactor out the constant from the second term.✓ Proved
- \[ = - \frac{\cos{\left(10 x \right)} \frac{d}{d x} 10 x}{20} + \frac{d}{d x} \frac{x}{2} \]trigApply the derivative rule for sine.✓ Proved
- \[ = - \frac{\cos{\left(10 x \right)}}{2} + \frac{d}{d x} \frac{x}{2} \]derivative algebra simplifyDifferentiate the inner function 10*x. Multiply the constants. Simplify the fraction.✓ Proved
- \[ = \frac{1}{2} - \frac{\cos{\left(10 x \right)}}{2} \]derivativeEvaluate the remaining derivative.✓ Proved
Answer \( \sin^{2}{\left(5 x \right)} \)
✓ 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. 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 |
| 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) — The stated answer sin(5*x)**2 is mathematically incorrect; the correct derivative is 1/2 - 1/2*cos(10*x). While the step-by-step derivation in the solution is correct, the final stated answer contradicts the result of the derivation.
Senior review claude-sonnet-5-5, 2026-10-13: pass — Each step is correct and uses an acceptable rule name, and the result 1/2 - cos(10x)/2 equals sin(5x)^2, so the stated answer is right. The panel's objection rests on a false belief that the two forms differ.
qwen3.6:27b-mlx: dismiss — The claim is false: 1/2 - cos(10x)/2 = sin(5x)^2 by the double-angle identity cos(10x) = 1 - 2 sin^2(5x), and SymPy proved the final line equals the stated answer.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-13 — Each step is correct and uses an acceptable rule name, and the result 1/2 - cos(10x)/2 equals sin(5x)^2, so the stated answer is right. The panel's objection rests on a false belief that the two forms differ.qwen3.6:27b-mlx: fail (error) 2026-09-27 — The stated answer sin(5*x)**2 is mathematically incorrect; the correct derivative is 1/2 - 1/2*cos(10*x). While the step-by-step derivation in the solution is correct, the final stated answer contradicts the result of the derivation.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 4 is labeled 'trig' but applies the chain rule to differentiate sin(10*x); the label 'trig' is not in the allowed vocabulary for this operation, and 'chain' is the correct label. Additionally, Step 4 applies two rules at once (derivative of sine and chain rule), violating the one-rule-per-step constraint.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 4 is labeled 'trig', but the step applies the chain rule to differentiate sin(10*x); the label 'chain' is required by the vocabulary for this operation. Additionally, Step 1 is unlabelled, which violates the requirement that each step names its rule.gpt-oss:20b: pass 2026-09-26
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 2026-09-27 with SymPy 1.14.0.