Derivative of \( \displaystyle - \frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4} \)
Problem 2.399 · hard Video
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4} \).
- \[ \frac{d}{d x} \left(- \frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \frac{\sin{\left(2 x \right)}}{4} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{\frac{d}{d x} \sin{\left(2 x \right)}}{4} \]constant-multipleFactor out the constant 1/4.✓ Proved
- \[ = \frac{\cos{\left(2 x \right)} \frac{d}{d x} 2 x}{4} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]chainApply the chain rule to sin(2*x).✓ Proved
- \[ = \frac{\cos{\left(2 x \right)}}{2} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]derivative algebraDifferentiate 2*x. Simplify the constant multiplication.✓ Proved
- \[ = \frac{\cos{\left(2 x \right)}}{2} - \frac{1}{2} \]derivative simplifyDifferentiate -x/2. Factor out 1/2.✓ 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 |
| 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: pass — The solution correctly applies the sum, constant-multiple, chain, and derivative rules in separate steps. The final result is algebraically equivalent to the stated answer.
Senior review claude-sonnet-5-5, 2026-10-08: pass — The derivative is -1/2 + cos(2x)/2 = (cos(2x) - 1)/2 = -sin(x)^2, which matches the stated answer. Every step changes one thing and its rule label is legitimate. SymPy proved every line, including the final comparison with the stated answer. The panel listed no specific objections to rule on.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-08 — The derivative is -1/2 + cos(2x)/2 = (cos(2x) - 1)/2 = -sin(x)^2, which matches the stated answer. Every step changes one thing and its rule label is legitimate. SymPy proved every line, including the final comparison with the stated answer. The panel listed no specific objections to rule on.qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the sum, constant-multiple, chain, and derivative rules in separate steps. The final result is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-20 — The final answer is incorrect; the derivative of f(x) is -1/2 + 1/2*cos(2x), which simplifies to -sin^2(x) only if the original function was integrated, not differentiated. The steps correctly compute the derivative as -1/2 + 1/2*cos(2x), but the stated answer claims it equals -sin(x)^2, which is mathematically false (e.g., at x=0, derivative is 0, but -sin^2(0)=0; at x=pi/2, derivative is -1, but -sin^2(pi/2)=-1; wait, -1/2 + 1/2*cos(pi) = -1/2 - 1/2 = -1. -sin^2(pi/2) = -1. Let's check x=pi/4. Deriv: -1/2 + 1/2*cos(pi/2) = -1/2. -sin^2(pi/4) = -1/2. It seems they are equal? Identity: cos(2x) = 1 - 2sin^2(x). So 1/2(cos(2x)-1) = 1/2(1-2sin^2(x)-1) = -sin^2(x). The math is actually correct. Why did I think it was wrong? Ah, I misread the task or the identity. Let me re-read carefully. f(x) = -x/2 + sin(2x)/4. f'(x) = -1/2 + 2cos(2x)/4 = -1/2 + cos(2x)/2. Identity: cos(2x) = 1 - 2sin^2(x). So f'(x) = -1/2 + (1 - 2sin^2(x))/2 = -1/2 + 1/2 - sin^2(x) = -sin^2(x). The math is correct. So why fail? Let's look at the steps again. Step 1 is unlabelled. The contract says 'Each step names its rule...'. Step 1 has [rule: unlabelled]. Is 'unlabelled' in the fixed vocabulary? No. The vocabulary is: algebra, chain, constant, constant-multiple, derivative, exponential, inverse-trig, logarithmic, power, product, quotient, rewrite, simplify, sum. 'unlabelled' is not in the list. This is a defect.gpt-oss:20b: pass 2026-09-20
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.