Derivative of \( \displaystyle \frac{3 x}{2} - \frac{\sin{\left(6 x + 2 \right)}}{4} \)
Problem 2.1250 · hard
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{\sin{\left(6 x + 2 \right)}}{4} \).
- \[ \frac{d}{d x} \left(\frac{3 x}{2} - \frac{\sin{\left(6 x + 2 \right)}}{4}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{3 x}{2} - \frac{d}{d x} \frac{\sin{\left(6 x + 2 \right)}}{4} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{3 x}{2} - \frac{\frac{d}{d x} \sin{\left(6 x + 2 \right)}}{4} \]constantFactor out the constant 1/4.✓ Proved
- \[ = - \frac{\cos{\left(6 x + 2 \right)} \frac{d}{d x} \left(6 x + 2\right)}{4} + \frac{d}{d x} \frac{3 x}{2} \]chainApply the chain rule to the sine term.✓ Proved
- \[ = - \frac{3 \cos{\left(6 x + 2 \right)}}{2} + \frac{d}{d x} \frac{3 x}{2} \]derivative algebraDifferentiate the inner function 6*x + 2. Simplify the constant coefficient.✓ Proved
- \[ = \frac{3}{2} - \frac{3 \cos{\left(6 x + 2 \right)}}{2} \]derivative simplifyDifferentiate the remaining term 3*x/2. Factor out the common term 3/2.✓ Proved
Answer \( \frac{3}{2} - \frac{3 \cos{\left(6 x + 2 \right)}}{2} \)
✓ 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.
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 |
| 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 the sum, constant multiple, chain, and derivative rules in separate steps. The final simplification is algebraically valid and clearly labeled.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the sum, constant multiple, chain, and derivative rules in separate steps. The final simplification is algebraically valid and clearly labeled.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (style) 2026-09-29 — Step 3 is labeled 'constant' but effectively applies the constant-multiple rule (factoring 1/4 out of the derivative). While 'constant' is in the vocabulary, 'constant-multiple' is the precise label for this operation, and using 'constant' is ambiguous or incorrect depending on strict interpretation (usually 'constant' refers to d/dx(c)=0). More critically, Step 8 changes the form of the answer from the stated answer `3/2 - 3*cos(6*x + 2)/2` to `3/2 * (1 - cos(6*x + 2))`. The task is to differentiate, and the stated answer is the standard simplified form. Step 8 is an unnecessary algebraic manipulation that deviates from the stated answer, and while labeled 'simplify', it makes the solution inconsistent with the provided 'Stated answer'. However, the primary defect is likely the labeling in Step 3 or the deviation in Step 8. Let's look closer. Step 3: `Derivative(sin(6*x + 2)/4, x)` becomes `(1/4)*Derivative(sin(6*x + 2), x)`. This is the constant multiple rule. The vocabulary has `constant-multiple`. The label used is `constant`. This is a labeling defect. Step 8: The stated answer is `3/2 - 3*cos(6*x + 2)/2`. The final step produces `3/2 * (1 - cos(6*x + 2))`. These are algebraically equal, but the solution ends with a different form than the stated answer. This is a style/consistency issue. The prompt says 'A computer algebra system has already confirmed that consecutive lines are algebraically equal'. So the math is fine. The issue is the label 'constant' for a constant-multiple step. Is 'constant' acceptable for constant-multiple? The instructions say: 'constant-multiple' is in the list. 'constant' is in the list. Usually 'constant' means derivative of a constant is 0. Using 'constant' for factoring out a constant is imprecise/wrong label. Also, Step 1 is 'unlabelled'. The instructions say 'Each step names its rule...'. Step 1 has no rule name. This is a defect.gpt-oss:20b: pass 2026-09-29
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 2026-09-29 with SymPy 1.14.0.