Derivative of \( \displaystyle - \frac{x}{2} + \frac{\sin{\left(10 x \right)}}{20} \)
Problem 2.1357 · hard
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(10 x \right)}}{20} \).
- \[ \frac{d}{d x} \left(- \frac{x}{2} + \frac{\sin{\left(10 x \right)}}{20}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \frac{\sin{\left(10 x \right)}}{20} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{\frac{d}{d x} \sin{\left(10 x \right)}}{20} \]constant-multiple constant-multipleFactor out the constant 1/20. Factor out the constant -1/2.✓ Proved
- \[ = - \frac{\frac{d}{d x} x}{2} + \frac{\frac{d}{d x} \sin{\left(10 x \right)}}{20} \]constant-multiple✓ Proved
- \[ = \frac{\frac{d}{d x} \sin{\left(10 x \right)}}{20} - \frac{1}{2} \]derivative simplifyDifferentiate x. Simplify the first term.✓ Proved
- \[ = \frac{\cos{\left(10 x \right)} \frac{d}{d x} 10 x}{20} - \frac{1}{2} \]trigDifferentiate sin(10*x) using the chain rule.✓ Proved
- \[ = \frac{\cos{\left(10 x \right)}}{2} - \frac{1}{2} \]chain algebra simplifyDifferentiate 10*x. Multiply the constants. Simplify the final expression.✓ Proved
Answer \( - \sin^{2}{\left(5 x \right)} \)
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. A reviewer's objection to the explanation stands; see the receipt.
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: fail (misleading) — Step 8 incorrectly labels the chain rule as "trig" and step 9 incorrectly labels the derivative of a linear function as "chain". The correct labels are: step 8 – "chain", step 9 – "derivative". This mislabeling could teach students the wrong rule names.qwen3.6:27b-mlx: fail (error) — The final result is mathematically incorrect; the derivative of -x/2 is -1/2, but the stated answer is -sin(5*x)^2. Additionally, Step 8 mislabels the application of the chain rule as 'trig'.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The final result is mathematically incorrect; the derivative of -x/2 is -1/2, but the stated answer is -sin(5*x)^2. Additionally, Step 8 mislabels the application of the chain rule as 'trig'.gpt-oss:20b: fail (misleading) 2026-10-03 — Step 8 incorrectly labels the chain rule as "trig" and step 9 incorrectly labels the derivative of a linear function as "chain". The correct labels are: step 8 – "chain", step 9 – "derivative". This mislabeling could teach students the wrong rule names.qwen3.6:27b-mlx: fail (error) 2026-09-30 — The final answer is mathematically incorrect; the derivative of -x/2 + sin(10x)/20 is -1/2 + cos(10x)/2, not -sin(5x)^2. Additionally, step 8 mislabels the application of the chain rule as 'trig'.gpt-oss:20b: fail (error) 2026-09-30 — Step 8 applies both the trig rule (derivative of sin) and the chain rule in a single step, violating the rule that each step must change only one thing. The label "trig" is also misleading because the chain rule is also applied here.
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.