Derivative of \( \displaystyle \frac{5 x}{2} - \frac{5 \sin{\left(4 x + 2 \right)}}{8} \)
Problem 2.691 · hard
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(4 x + 2 \right)}}{8} \).
- \[ \frac{d}{d x} \left(\frac{5 x}{2} - \frac{5 \sin{\left(4 x + 2 \right)}}{8}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{d}{d x} \frac{5 \sin{\left(4 x + 2 \right)}}{8} \]sum constant-multipleApply the sum rule. Factor out the constants.✓ Proved
- \[ = \frac{5 \frac{d}{d x} x}{2} - \frac{5 \frac{d}{d x} \sin{\left(4 x + 2 \right)}}{8} \]constant-multipleApply the constant multiple rule to both terms.✓ Proved
- \[ = \frac{5}{2} - \frac{5 \frac{d}{d x} \sin{\left(4 x + 2 \right)}}{8} \]derivative algebraDifferentiate x. Simplify the first term.✓ Proved
- \[ = - \frac{5 \cos{\left(4 x + 2 \right)} \frac{d}{d x} \left(4 x + 2\right)}{8} + \frac{5}{2} \]trigApply the derivative of sine.✓ Proved
- \[ = - \frac{5 \cos{\left(4 x + 2 \right)} \frac{d}{d x} x}{2} + \frac{5}{2} \]chainApply the chain rule.✓ Proved
- \[ = \frac{5}{2} - \frac{5 \cos{\left(4 x + 2 \right)}}{2} \]derivative algebra simplifyDifferentiate 4*x + 2. Simplify the expression. Simplify the fraction 20/8 to 5/2.✓ Proved
Answer \( \frac{5}{2} - \frac{5 \cos{\left(4 x + 2 \right)}}{2} \)
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 |
| 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: passqwen3.6:27b-mlx: fail (style) — Step 3 applies the constant-multiple rule to both terms simultaneously, violating the one-change-per-step constraint. Step 8 incorrectly labels the expansion of the inner derivative as 'chain'; the chain rule was already applied in Step 7, and Step 8 is merely expanding the derivative of the inner function.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 3 applies the constant-multiple rule to both terms simultaneously, violating the one-change-per-step constraint. Step 8 incorrectly labels the expansion of the inner derivative as 'chain'; the chain rule was already applied in Step 7, and Step 8 is merely expanding the derivative of the inner function.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (error) 2026-09-21 — Step 8 applies two rules at once: it uses the chain rule to bring down the inner derivative and then applies the derivative of the linear function 4*x+2, effectively combining chain and derivative in one step. This violates the rule that each step must change only one thing.
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.