Derivative of \( \displaystyle \frac{5 x}{2} - \frac{\sin{\left(10 x \right)}}{4} \)
Problem 2.1378 · hard
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{\sin{\left(10 x \right)}}{4} \).
- \[ \frac{d}{d x} \left(\frac{5 x}{2} - \frac{\sin{\left(10 x \right)}}{4}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{d}{d x} \frac{\sin{\left(10 x \right)}}{4} \]sum constantApply the sum rule. Factor out the constants.✓ Proved
- \[ = \frac{5 \frac{d}{d x} x}{2} - \frac{\frac{d}{d x} \sin{\left(10 x \right)}}{4} \]constant-multipleApply the constant multiple rule.✓ Proved
- \[ = \frac{5}{2} - \frac{\frac{d}{d x} \sin{\left(10 x \right)}}{4} \]derivative algebraDifferentiate x. Simplify the first term.✓ Proved
- \[ = - \frac{\cos{\left(10 x \right)} \frac{d}{d x} 10 x}{4} + \frac{5}{2} \]chainApply the chain rule to sin(10*x).✓ Proved
- \[ = \frac{5}{2} - \frac{5 \cos{\left(10 x \right)}}{2} \]derivative algebra simplifyDifferentiate 10*x. Multiply the constants. Simplify the fraction.✓ Proved
Answer \( 5 \sin^{2}{\left(5 x \right)} \)
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. The reviewers disagree about how one step is explained; every verdict is in 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 |
| 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 differentiation rules step-by-step. The final answer is algebraically equivalent to the stated answer (using the identity 1 - cos(2u) = 2sin^2(u)).
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules step-by-step. The final answer is algebraically equivalent to the stated answer (using the identity 1 - cos(2u) = 2sin^2(u)).gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — 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. The final result is algebraically equivalent to the stated answer.gpt-oss:20b: fail (error) 2026-10-03 — The final answer 5*sin(5*x)**2 does not match the correct derivative 5/2 - (5/2)cos(10*x). Additionally, step 1 uses an unlabelled rule instead of the required "derivative" label.
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.