Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(6 x \right)}}{12} \)
Problem 2.684 · hard Video
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(6 x \right)}}{12} \).
- \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(6 x \right)}}{12}\right) \]sumStart with the derivative of the entire function.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(6 x \right)}}{12} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(6 x \right)}}{12} \]constantFactor out the constant 1/12.✓ Proved
- \[ = \frac{\frac{d}{d x} x}{2} - \frac{\frac{d}{d x} \sin{\left(6 x \right)}}{12} \]constant-multipleFactor out the constant 1/2.✓ Proved
- \[ = \frac{1}{2} - \frac{\frac{d}{d x} \sin{\left(6 x \right)}}{12} \]derivativeDifferentiate the first term.✓ Proved
- \[ = - \frac{\cos{\left(6 x \right)} \frac{d}{d x} 6 x}{12} + \frac{1}{2} \]trigApply the chain rule to the sine term.✓ Proved
- \[ = \frac{1}{2} - \frac{\cos{\left(6 x \right)}}{2} \]derivative algebra simplifyDifferentiate the inner function 6*x. Multiply the constants. Simplify the fraction.✓ Proved
Answer \( \sin^{2}{\left(3 x \right)} \)
✓ 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 |
| 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 (style) — Step 6 applies two rules at once (the derivative of sin and the chain rule) and is labeled incorrectly as "trig". The correct label should be "chain" (or a combination of "derivative" and "chain" applied separately).qwen3.6:27b-mlx: fail (style) — Step 6 is labeled 'trig' but applies the chain rule; the label should be 'chain'. Step 3 is labeled 'constant' but factors out a constant multiplier, which is better described as 'constant-multiple'.
Senior review claude-sonnet-5-5, 2026-10-11: pass — The derivative 1/2 - cos(6x)/2 is correct and equals sin(3x)^2 by the half-angle identity, which SymPy confirmed on the final line. Every step is a valid single change with an acceptable label.
gpt-oss:20b: dismiss — Step 6 is a single chain-rule step that uses the sine derivative, so either 'trig' or 'chain' is an acceptable label, and it does not do two independent things.qwen3.6:27b-mlx: dismiss — Both 'trig' and 'chain' legitimately describe step 6, and 'constant' is an acceptable label for pulling the constant 1/12 out of the derivative in step 3, so neither is a wrong rule name.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-11 — The derivative 1/2 - cos(6x)/2 is correct and equals sin(3x)^2 by the half-angle identity, which SymPy confirmed on the final line. Every step is a valid single change with an acceptable label.qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 6 is labeled 'trig' but applies the chain rule; the label should be 'chain'. Step 3 is labeled 'constant' but factors out a constant multiplier, which is better described as 'constant-multiple'.gpt-oss:20b: fail (style) 2026-09-26 — Step 6 applies two rules at once (the derivative of sin and the chain rule) and is labeled incorrectly as "trig". The correct label should be "chain" (or a combination of "derivative" and "chain" applied separately).qwen3.6:27b-mlx: fail (misleading) 2026-09-21 — Step 6 is labeled 'trig' but applies the chain rule; the label should be 'chain'. Additionally, the stated answer sin(3*x)**2 is not algebraically equal to the derived result 1/2 - 1/2*cos(6*x) in the context of the derivative (the derivative is 1/2 - 1/2*cos(6*x), which equals sin^2(3x), but the problem asks for the derivative, and while the final expression is correct, the step labels are the primary defect here. Wait, sin^2(3x) = (1-cos(6x))/2 = 1/2 - 1/2 cos(6x). So the final answer is correct. The defect is the label in step 6.gpt-oss:20b: fail (error) 2026-09-21 — Step 6 applies two rules at once: it differentiates sin(6*x) using the chain rule, which involves both the derivative of sin (a trig rule) and the derivative of the inner function 6*x (a derivative rule). The label "trig" is also incorrect for this step. Additionally, step 6 should be split into two separate steps to comply with the one‑change rule.qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 6 is labeled 'trig' but applies the chain rule; the label should be 'chain'. Step 3 is labeled 'constant' but factors out a constant multiplier, which is better described as 'constant-multiple'.gpt-oss:20b: fail (error) 2026-09-21 — Step 6 applies two rules at once (chain rule and the derivative of sin) and is labeled incorrectly as "trig". This violates the one‑rule‑per‑step rule and mislabels the operation.
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.