Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(10 x - 2 \right)}}{20} \)
Problem 2.69 · hard Video
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
- \[ = - \frac{\cos{\left(10 x - 2 \right)} \frac{d}{d x} \left(10 x - 2\right)}{20} + \frac{d}{d x} \frac{x}{2} \]chainApply the chain rule to the sine term.✓ Proved
- \[ = - \frac{\cos{\left(10 x - 2 \right)} \frac{d}{d x} x}{2} + \frac{d}{d x} \frac{x}{2} \]derivative algebra simplifyDifferentiate the inner function 10*x - 2. Multiply the constants. Simplify the fraction 10/20.✓ Proved
- \[ = - \frac{\cos{\left(10 x - 2 \right)}}{2} + \frac{d}{d x} \frac{x}{2} \]derivativeDifferentiate x with respect to x.✓ Proved
- \[ = \frac{1}{2} - \frac{\cos{\left(10 x - 2 \right)}}{2} \]simplifyFinal simplified expression.✓ 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. 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 |
| 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 (error) — Step 3 is labeled 'constant' but applies the constant-multiple rule to the second term while leaving the first term untouched. The label 'constant' typically refers to the derivative of a constant being zero, whereas 'constant-multiple' is the correct label for factoring out a coefficient. Furthermore, Step 5 is labeled 'derivative' but performs algebraic simplification of the inner function's derivative (10x-2 -> 10), which should be labeled 'algebra' or 'simplify', while the actual differentiation of the linear term happens in Step 8. The labeling is inconsistent and incorrect regarding the specific rules applied.
Senior review claude-sonnet-5-5, 2026-10-03: fail — The math is correct and SymPy proved every line and the final answer. Step 3 should be labeled 'constant-multiple' instead of 'constant'; the Step 5 label is acceptable.
qwen3.6:27b-mlx: uphold — The Step 3 objection is right: pulling the factor 1/20 out of the second term is the constant-multiple rule, and 'constant' names the derivative-of-a-constant rule, which was not applied. The Step 5 objection is wrong, because Step 5 evaluates the derivative of the inner function 10*x - 2, which is a legitimate 'derivative' step.
Every verdict on record (13)
claude-sonnet-5-5: fail (style) 2026-10-03 — The math is correct and SymPy proved every line and the final answer. Step 3 should be labeled 'constant-multiple' instead of 'constant'; the Step 5 label is acceptable.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 is labeled 'constant' but applies the constant-multiple rule to the second term while leaving the first term untouched. The label 'constant' typically refers to the derivative of a constant being zero, whereas 'constant-multiple' is the correct label for factoring out a coefficient. Furthermore, Step 5 is labeled 'derivative' but performs algebraic simplification of the inner function's derivative (10x-2 -> 10), which should be labeled 'algebra' or 'simplify', while the actual differentiation of the linear term happens in Step 8. The labeling is inconsistent and incorrect regarding the specific rules applied.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 3 incorrectly labels the extraction of the constant factor 1/20 as 'constant'. The rule 'constant' applies to the derivative of a constant (which is zero), whereas extracting a constant multiplier requires the 'constant-multiple' rule.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 3 labels the extraction of the constant factor 1/20 as 'constant', but the vocabulary specifies 'constant-multiple' for this rule. Additionally, Step 5 applies the derivative to the inner function but leaves the outer derivative operator on the first term, which is a mixed operation not strictly covered by a single 'derivative' label in the context of the whole expression, though the primary defect is the mislabeling in Step 3.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.