Derivative of \( \displaystyle - \frac{x}{2} + \frac{\sin{\left(10 x + 4 \right)}}{20} \)
Problem 2.1403 · hard
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(10 x + 4 \right)}}{20} \).
- \[ \frac{d}{d x} \left(- \frac{x}{2} + \frac{\sin{\left(10 x + 4 \right)}}{20}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \frac{\sin{\left(10 x + 4 \right)}}{20} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{\frac{d}{d x} \sin{\left(10 x + 4 \right)}}{20} \]constantPull out the constant factor.✓ Proved
- \[ = \frac{\cos{\left(10 x + 4 \right)} \frac{d}{d x} \left(10 x + 4\right)}{20} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]chainApply the chain rule to the sine function.✓ Proved
- \[ = \frac{\cos{\left(10 x + 4 \right)}}{2} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]derivativeDifferentiate the inner function 10*x + 4.✓ Proved
- \[ = \frac{\cos{\left(10 x + 4 \right)}}{2} - \frac{1}{2} \]derivative algebra simplifyDifferentiate the first term. Multiply the constants. Simplify the fraction.✓ Proved
Answer \( \frac{\cos{\left(10 x + 4 \right)} - 1}{2} \)
✓ 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.
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 rule, constant multiple rule, and chain rule in separate steps. The final simplification is algebraically correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the sum rule, constant multiple rule, and chain rule in separate steps. The final simplification is algebraically correct.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03
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.