Derivative of \( \displaystyle - \frac{x}{2} + \frac{\sin{\left(6 x + 4 \right)}}{12} \)
Problem 2.696 · hard Video
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(6 x + 4 \right)}}{12} \).
- \[ \frac{d}{d x} \left(- \frac{x}{2} + \frac{\sin{\left(6 x + 4 \right)}}{12}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \frac{\sin{\left(6 x + 4 \right)}}{12} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{\frac{d}{d x} \sin{\left(6 x + 4 \right)}}{12} \]constantPull out the constant factor.✓ Proved
- \[ = \frac{\cos{\left(6 x + 4 \right)} \frac{d}{d x} \left(6 x + 4\right)}{12} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]chainApply the chain rule to the sine function.✓ Proved
- \[ = \frac{\cos{\left(6 x + 4 \right)}}{2} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]derivative algebraDifferentiate the inner function 6*x + 4. Simplify the constant multiplication.✓ Proved
- \[ = \frac{\cos{\left(6 x + 4 \right)}}{2} - \frac{1}{2} \]derivativeEvaluate the derivative of the first term.✓ Proved
Answer \( \frac{\cos{\left(6 x + 4 \right)} - 1}{2} \)
✓ 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 |
| 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 rule, constant multiple rule, and chain rule in separate steps. All labels are valid and the algebraic simplifications are correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the sum rule, constant multiple rule, and chain rule in separate steps. All labels are valid and the algebraic simplifications are correct.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the sum, constant multiple, chain, and derivative rules in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-21
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.