Derivative of \( \displaystyle \frac{3 x}{2} - \frac{\sin{\left(6 x - 6 \right)}}{4} \)
Problem 2.1745 · hard
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{\sin{\left(6 x - 6 \right)}}{4} \).
- \[ \frac{d}{d x} \left(\frac{3 x}{2} - \frac{\sin{\left(6 x - 6 \right)}}{4}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{3 x}{2} - \frac{d}{d x} \frac{\sin{\left(6 x - 6 \right)}}{4} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{3 x}{2} - \frac{\frac{d}{d x} \sin{\left(6 x - 6 \right)}}{4} \]constantFactor out the constant 1/4.✓ Proved
- \[ = - \frac{\cos{\left(6 x - 6 \right)} \frac{d}{d x} \left(6 x - 6\right)}{4} + \frac{d}{d x} \frac{3 x}{2} \]chainApply the chain rule to the sine term.✓ Proved
- \[ = - \frac{3 \cos{\left(6 x - 6 \right)} \frac{d}{d x} x}{2} + \frac{d}{d x} \frac{3 x}{2} \]chain constant-multipleApply the chain rule to the inner linear function. Multiply the constants.✓ Proved
- \[ = - \frac{3 \cos{\left(6 x - 6 \right)} \frac{d}{d x} x}{2} + \frac{3 \frac{d}{d x} x}{2} \]powerDifferentiate the first term.✓ Proved
- \[ = \frac{3}{2} - \frac{3 \cos{\left(6 x - 6 \right)}}{2} \]derivative simplifyDifferentiate the remaining x terms. Simplify the expression.✓ Proved
Answer \( \frac{3}{2} - \frac{3 \cos{\left(6 x - 6 \right)}}{2} \)
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 |
| 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 5 incorrectly applies the chain rule to the linear inner function; the derivative of 6*x‑6 is 6, not 6*Derivative(x,x). Additionally, step 7 labels the differentiation of 3*x/2 as a “power” rule, which is inappropriate.qwen3.6:27b-mlx: fail (style) — Step 3 uses the label 'constant' to factor out 1/4, but the vocabulary requires 'constant-multiple' for this operation. Step 7 uses the label 'power' to differentiate 3x/2, but 'derivative' is the correct label for unfolding d/dx on a linear term.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — Step 5 incorrectly applies the chain rule to the linear inner function; the derivative of 6*x‑6 is 6, not 6*Derivative(x,x). Additionally, step 7 labels the differentiation of 3*x/2 as a “power” rule, which is inappropriate.qwen3.6:27b-mlx: fail (style) 2026-10-07 — Step 3 uses the label 'constant' to factor out 1/4, but the vocabulary requires 'constant-multiple' for this operation. Step 7 uses the label 'power' to differentiate 3x/2, but 'derivative' is the correct label for unfolding d/dx on a linear term.gpt-oss:20b: fail (style) 2026-10-07 — Step 7 incorrectly labels the differentiation of (3/2)*x as a "power" rule; it should be a "constant-multiple" (or simply "derivative") step. The rest of the steps are fine.qwen3.6:27b-mlx: fail (style) 2026-10-07 — Step 3 is labeled 'constant' but factors out a constant from a derivative, which should be labeled 'constant-multiple'. Step 7 is labeled 'power' but performs the differentiation of a linear term (which is a 'derivative' or 'constant-multiple' step), not a power rule application.
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 2026-10-07 with SymPy 1.14.0.