Derivative of \( \displaystyle \frac{5 x}{2} - \frac{5 \sin{\left(2 x \right)}}{4} \)
Problem 2.281 · hard Beautiful Video
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(2 x \right)}}{4} \).
- \[ \frac{d}{d x} \left(\frac{5 x}{2} - \frac{5 \sin{\left(2 x \right)}}{4}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{d}{d x} \frac{5 \sin{\left(2 x \right)}}{4} \]sum constant-multipleApply the sum rule. Factor out the constants.✓ Proved
- \[ = \frac{5 \frac{d}{d x} x}{2} - \frac{5 \frac{d}{d x} \sin{\left(2 x \right)}}{4} \]constant-multipleSeparate the constants from the derivatives.✓ Proved
- \[ = \frac{5}{2} - \frac{5 \frac{d}{d x} \sin{\left(2 x \right)}}{4} \]derivative algebraDifferentiate x. Simplify the first term.✓ Proved
- \[ = - \frac{5 \cos{\left(2 x \right)} \frac{d}{d x} 2 x}{4} + \frac{5}{2} \]chainApply the chain rule to sin(2*x).✓ Proved
- \[ = \frac{5}{2} - \frac{5 \cos{\left(2 x \right)}}{2} \]derivative algebraDifferentiate 2*x. Multiply the constants.✓ Proved
Answer \( 5 \sin^{2}{\left(x \right)} \)
✓ 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: fail (misleading) — The solution does not show how the derivative simplifies to 5 sin²(x), omitting a crucial step that would help students understand the process.qwen3.6:27b-mlx: pass — 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.
Senior review claude-sonnet-5-5, 2026-10-06: pass — Every line is an equivalent form and SymPy proved each step. The derivative 5/2 - 5/2*cos(2x) equals 5*sin(x)**2 by the half-angle identity. Line 3 only rewrites 5*x/2 as 5/2*x, so its constant-multiple label is slightly loose. The real constant-multiple step is line 4, so this is at most a style point.
agent:pipeline: uphold — Beautiful nomination: A messy-looking function (a linear term minus a scaled sin(2x)) differentiates to 5/2*(1 - cos 2x), which collapses by the half-angle identity to 5*sin(x)**2.
Every verdict on record (13)
claude-sonnet-5-5: pass 2026-10-06 — Every line is an equivalent form and SymPy proved each step. The derivative 5/2 - 5/2*cos(2x) equals 5*sin(x)**2 by the half-angle identity. Line 3 only rewrites 5*x/2 as 5/2*x, so its constant-multiple label is slightly loose. The real constant-multiple step is line 4, so this is at most a style point.qwen3.6:27b-mlx: pass 2026-09-20 — 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 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the sum, constant-multiple, derivative, and chain rules in a step-by-step manner. The final result is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-19 — The final answer provided (5*sin(x)**2) does not match the result of the step-by-step derivation (5/2 - 5/2*cos(2*x)). Although these expressions are equivalent via trigonometric identities, the solution fails to include a step transforming the derived expression into the stated answer, making the final claim unsupported by the preceding work.deepseek-r1:70b: fail (misleading) 2026-09-19 — The solution does not show how the derivative simplifies to 5 sin²(x), omitting a crucial step that would help students understand the process.gpt-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 vocabulary.deepseek-r1:70b: fail (error) 2026-09-19 — The final answer is incorrect; the derivative should be 5/2 - (5/2)cos(2x), not 5sin²(x).gpt-oss:20b: pass 2026-09-19gpt-oss:20b: pass 2026-09-17deepseek-r1:70b: pass 2026-09-17 — The solution correctly applies differentiation rules and simplifies appropriately. The final form using the trigonometric identity is valid, even though not explicitly shown.
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.