Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(10 x + 2 \right)}}{20} \)
Problem 2.41 · hard
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(10 x + 2 \right)}}{20} \).
- \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(10 x + 2 \right)}}{20}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(10 x + 2 \right)}}{20} \]sumApply the sum rule for differentiation.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(10 x + 2 \right)}}{20} \]constantFactor out the constant 1/20.✓ Proved
- \[ = - \frac{\cos{\left(10 x + 2 \right)} \frac{d}{d x} \left(10 x + 2\right)}{20} + \frac{d}{d x} \frac{x}{2} \]trigApply the derivative rule for sine.✓ Proved
- \[ = - \frac{\cos{\left(10 x + 2 \right)} \frac{d}{d x} 2}{20} - \frac{\cos{\left(10 x + 2 \right)} \frac{d}{d x} 10 x}{20} + \frac{d}{d x} \frac{x}{2} \]sumApply the sum rule to the inner function.✓ Proved
- \[ = - \frac{\cos{\left(10 x + 2 \right)}}{2} + \frac{d}{d x} \frac{x}{2} \]derivative algebra simplifyDifferentiate the terms inside the chain rule. Multiply the constants. Simplify the fraction.✓ Proved
- \[ = \frac{1}{2} - \frac{\cos{\left(10 x + 2 \right)}}{2} \]derivative algebraEvaluate the remaining derivative. Factor out 1/2 for the final simplified form.✓ Proved
Answer \( \frac{1}{2} - \frac{\cos{\left(10 x + 2 \right)}}{2} \)
✓ 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.
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 |
| 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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: fail (style) — Step 3 applies the constant-multiple rule, but is labeled 'constant'. Step 5 applies the sum rule to the inner derivative, but the label 'sum' is acceptable; however, the primary defect is Step 3's label which does not match the fixed vocabulary for factoring out a constant multiplier (which is 'constant-multiple').
Every verdict on record (12)
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 3 applies the constant-multiple rule, but is labeled 'constant'. Step 5 applies the sum rule to the inner derivative, but the label 'sum' is acceptable; however, the primary defect is Step 3's label which does not match the fixed vocabulary for factoring out a constant multiplier (which is 'constant-multiple').gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 3 applies the constant-multiple rule, but uses the label 'constant', which is not in the fixed vocabulary. Step 5 applies the sum rule to the argument of the derivative, but the label 'sum' is acceptable; however, the step structure is slightly ambiguous as it expands the derivative of a sum inside a product context, though technically valid under the sum rule. The primary defect is the invalid label 'constant' in step 3.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19 — 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.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: pass 2026-09-17deepseek-r1:70b: pass 2026-09-17
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.