Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(6 x - 2 \right)}}{12} \)
Problem 2.1126 · hard Video
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(6 x - 2 \right)}}{12} \).
- \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(6 x - 2 \right)}}{12}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(6 x - 2 \right)}}{12} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(6 x - 2 \right)}}{12} \]constantFactor out the constant 1/12.✓ Proved
- \[ = - \frac{\cos{\left(6 x - 2 \right)} \frac{d}{d x} \left(6 x - 2\right)}{12} + \frac{d}{d x} \frac{x}{2} \]chainApply the chain rule to the sine term.✓ Proved
- \[ = - \frac{\cos{\left(6 x - 2 \right)} \frac{d}{d x} x}{2} + \frac{d}{d x} \frac{x}{2} \]algebra algebraSimplify the product of constants. Simplify 6/12 to 1/2.✓ Proved
- \[ = - \frac{\cos{\left(6 x - 2 \right)}}{2} + \frac{d}{d x} \frac{x}{2} \]derivativeDifferentiate the remaining x term.✓ Proved
- \[ = \frac{1}{2} - \frac{\cos{\left(6 x - 2 \right)}}{2} \]simplifyFinal simplification.✓ Proved
Answer \( \frac{1}{2} - \frac{\cos{\left(6 x - 2 \right)}}{2} \)
✓ 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 (style) — Step 3 uses the label 'constant' but performs a constant-multiple extraction; the vocabulary requires 'constant-multiple' for this operation. Additionally, Step 5 applies algebraic simplification to constants while leaving a derivative operator intact, which is a presentational step better labeled 'simplify' or 'algebra', but the primary defect is the mislabeling in Step 3.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-03 — Step 3 uses the label 'constant' but performs a constant-multiple extraction; the vocabulary requires 'constant-multiple' for this operation. Additionally, Step 5 applies algebraic simplification to constants while leaving a derivative operator intact, which is a presentational step better labeled 'simplify' or 'algebra', but the primary defect is the mislabeling in Step 3.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-28gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (style) 2026-09-28 — Step 3 is labeled 'constant' but factors out a constant from a derivative, which corresponds to the 'constant-multiple' rule in the provided vocabulary. Step 5 is labeled 'algebra' but applies the chain rule to differentiate the inner function (6x-2), which should be labeled 'chain' or split into separate steps.gpt-oss:20b: pass 2026-09-28
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 2026-10-03 with SymPy 1.14.0.