Derivative of \( \displaystyle - \frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4} \)
Problem 2.399 · hard
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 \( - \sin^{2}{\left(x \right)} \)
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 |
| 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.
Every verdict on record (6)
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.