Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(10 x - 6 \right)}}{20} \)
Problem 2.550 · hard Video
- \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(10 x - 6 \right)}}{20}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(10 x - 6 \right)}}{20} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(10 x - 6 \right)}}{20} \]constantPull out the constant factor.✓ Proved
- \[ = - \frac{\cos{\left(10 x - 6 \right)} \frac{d}{d x} \left(10 x - 6\right)}{20} + \frac{d}{d x} \frac{x}{2} \]chainApply the chain rule to the sine function.✓ Proved
- \[ = - \frac{\left(- \frac{d}{d x} 6 + \frac{d}{d x} 10 x\right) \cos{\left(10 x - 6 \right)}}{20} + \frac{d}{d x} \frac{x}{2} \]sumDifferentiate the inner function.✓ Proved
- \[ = - \frac{\cos{\left(10 x - 6 \right)} \frac{d}{d x} 10 x}{20} + \frac{d}{d x} \frac{x}{2} \]constantThe derivative of the constant 6 is 0.✓ Proved
- \[ = - \frac{\cos{\left(10 x - 6 \right)} \frac{d}{d x} x}{2} + \frac{d}{d x} \frac{x}{2} \]constant algebra simplifyThe derivative of 10*x is 10. Multiply the constants. Simplify the fraction 10/20.✓ Proved
- \[ = \frac{1}{2} - \frac{\cos{\left(10 x - 6 \right)}}{2} \]derivative simplifyEvaluate the derivatives of the remaining terms. Simplify the final expression.✓ Proved
✓ 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 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 11 | ✓ 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 a logical sequence. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
Senior review claude-sonnet-5-5, 2026-10-10: pass — The derivative 1/2 - 10cos(10x-6)/20 = 1/2 - cos(10x-6)/2 is correct, and every line was proved equal to the previous one. The only quibble is that 'constant' is used for pulling out a constant factor (lines 3 and 7), where 'constant-multiple' would be more precise. This is minor and arguably acceptable, so it does not justify a failure.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-10 — The derivative 1/2 - 10cos(10x-6)/20 = 1/2 - cos(10x-6)/2 is correct, and every line was proved equal to the previous one. The only quibble is that 'constant' is used for pulling out a constant factor (lines 3 and 7), where 'constant-multiple' would be more precise. This is minor and arguably acceptable, so it does not justify a failure.qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the sum, constant multiple, chain, and derivative rules in a logical sequence. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint and using valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 7 applies two rules at once: it pulls out the constant 10 (constant-multiple) and evaluates Derivative(x, x) to 1 (derivative), violating the one-rule-per-step constraint. Step 10 also combines evaluating Derivative(x/2, x) and Derivative(x, x) into a single step.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.