Derivative of \( \displaystyle \frac{5 x}{2} - \frac{5 \sin{\left(2 x + 4 \right)}}{4} \)
Problem 2.888 · hard Video
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(2 x + 4 \right)}}{4} \).
- \[ \frac{d}{d x} \left(\frac{5 x}{2} - \frac{5 \sin{\left(2 x + 4 \right)}}{4}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{d}{d x} \frac{5 \sin{\left(2 x + 4 \right)}}{4} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{5 \frac{d}{d x} \sin{\left(2 x + 4 \right)}}{4} \]constant-multipleFactor out the constant from the second term.✓ Proved
- \[ = \frac{5 \frac{d}{d x} x}{2} - \frac{5 \frac{d}{d x} \sin{\left(2 x + 4 \right)}}{4} \]constant-multipleFactor out the constant from the first term.✓ Proved
- \[ = \frac{5}{2} - \frac{5 \frac{d}{d x} \sin{\left(2 x + 4 \right)}}{4} \]derivative simplifyDifferentiate x. Simplify the first term.✓ Proved
- \[ = - \frac{5 \cos{\left(2 x + 4 \right)} \frac{d}{d x} \left(2 x + 4\right)}{4} + \frac{5}{2} \]chainApply the chain rule to the sine function.✓ Proved
- \[ = \frac{5}{2} - \frac{5 \cos{\left(2 x + 4 \right)}}{2} \]derivative algebra simplifyDifferentiate the inner function 2*x + 4. Multiply the constants. Factor out 5/2.✓ Proved
Answer \( \frac{5}{2} - \frac{5 \cos{\left(2 x + 4 \right)}}{2} \)
✓ 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 |
| 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: fail (style) — Step 10 factors out a constant to change the form of the answer, which is not required by the task and deviates from the stated answer. While not mathematically wrong, it is an unnecessary step that obscures the direct result.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-09-27 — Step 10 factors out a constant to change the form of the answer, which is not required by the task and deviates from the stated answer. While not mathematically wrong, it is an unnecessary step that obscures the direct result.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 10 factors out 5/2 to produce a form that differs from the stated answer (5/2 - 5/2 * cos(2*x + 4)). The solution should stop at Step 9 to match the stated answer, or the stated answer should be updated. Additionally, Step 1 is labeled 'unlabelled' which is not in the fixed vocabulary, though this is a minor formatting issue compared to the mismatch.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — 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 2026-09-26
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 2026-09-27 with SymPy 1.14.0.