Derivative of \( \displaystyle \frac{5 x}{2} - \frac{5 \sin{\left(6 x + 4 \right)}}{12} \)
Problem 2.1111 · hard
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(6 x + 4 \right)}}{12} \).
- \[ \frac{d}{d x} \left(\frac{5 x}{2} - \frac{5 \sin{\left(6 x + 4 \right)}}{12}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{d}{d x} \frac{5 \sin{\left(6 x + 4 \right)}}{12} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{5 \frac{d}{d x} \sin{\left(6 x + 4 \right)}}{12} \]constant-multipleFactor out the constant from the second term.✓ Proved
- \[ = \frac{5 \frac{d}{d x} x}{2} - \frac{5 \frac{d}{d x} \sin{\left(6 x + 4 \right)}}{12} \]constant-multipleFactor out the constant from the first term.✓ Proved
- \[ = \frac{5}{2} - \frac{5 \frac{d}{d x} \sin{\left(6 x + 4 \right)}}{12} \]derivativeDifferentiate x.✓ Proved
- \[ = - \frac{5 \cos{\left(6 x + 4 \right)} \frac{d}{d x} \left(6 x + 4\right)}{12} + \frac{5}{2} \]trigApply the derivative of sine.✓ Proved
- \[ = - \frac{5 \cos{\left(6 x + 4 \right)} \frac{d}{d x} 4}{12} - \frac{5 \cos{\left(6 x + 4 \right)} \frac{d}{d x} x}{2} + \frac{5}{2} \]chainApply the chain rule to the inner function.✓ Proved
- \[ = \frac{5}{2} - \frac{5 \cos{\left(6 x + 4 \right)}}{2} \]derivative algebra simplifyDifferentiate the components of the chain rule. Simplify the expression. Simplify the coefficients.✓ Proved
Answer \( \frac{5}{2} - \frac{5 \cos{\left(6 x + 4 \right)}}{2} \)
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 |
| 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 (error) — Step 7 applies both the chain rule and a constant‑multiple rule in a single step, violating the rule that each step must change only one thing.qwen3.6:27b-mlx: fail (style) — Step 7 applies the chain rule but also expands the derivative of the inner function into a sum of derivatives (derivative of 6x + derivative of 4), violating the one-rule-per-step constraint. This combines the chain rule application with the sum rule and constant multiple rule.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-28 — Step 7 applies the chain rule but also expands the derivative of the inner function into a sum of derivatives (derivative of 6x + derivative of 4), violating the one-rule-per-step constraint. This combines the chain rule application with the sum rule and constant multiple rule.gpt-oss:20b: fail (error) 2026-09-28 — Step 7 applies both the chain rule and a constant‑multiple rule in a single step, violating the rule that each step must change only one thing.qwen3.6:27b-mlx: pass 2026-09-28 — 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: fail (error) 2026-09-28 — Step 7 applies the chain rule twice and incorrectly differentiates the inner function of the sine. It expands cos(6*x+4) as if it were the outer function, producing an extra term involving Derivative(4,x).
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 2026-09-28 with SymPy 1.14.0.