Derivative of \( \displaystyle \frac{3 x}{2} - \frac{3 \sin{\left(2 x + 2 \right)}}{4} \)
Problem 2.1017 · hard
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(2 x + 2 \right)}}{4} \).
- \[ \frac{d}{d x} \left(\frac{3 x}{2} - \frac{3 \sin{\left(2 x + 2 \right)}}{4}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{3 x}{2} - \frac{d}{d x} \frac{3 \sin{\left(2 x + 2 \right)}}{4} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{3 x}{2} - \frac{3 \frac{d}{d x} \sin{\left(2 x + 2 \right)}}{4} \]constant-multiple constant-multipleFactor out the constant 3/4. Rewrite 3/2 as a constant multiplier.✓ Proved
- \[ = \frac{3 \frac{d}{d x} x}{2} - \frac{3 \frac{d}{d x} \sin{\left(2 x + 2 \right)}}{4} \]constant-multipleFactor out 3/2 from the first term.✓ Proved
- \[ = \frac{3}{2} - \frac{3 \frac{d}{d x} \sin{\left(2 x + 2 \right)}}{4} \]derivative simplifyDifferentiate x with respect to x. Simplify the first term.✓ Proved
- \[ = - \frac{3 \cos{\left(2 x + 2 \right)} \frac{d}{d x} \left(2 x + 2\right)}{4} + \frac{3}{2} \]trigApply the derivative rule for sine.✓ Proved
- \[ = - \frac{3 \cos{\left(2 x + 2 \right)} \frac{d}{d x} 2 x}{4} + \frac{3}{2} \]constantDifferentiate the constant 2.✓ Proved
- \[ = - \frac{3 \cos{\left(2 x + 2 \right)} \frac{d}{d x} x}{2} + \frac{3}{2} \]constant-multipleFactor out the constant 2.✓ Proved
- \[ = \frac{3}{2} - \frac{3 \cos{\left(2 x + 2 \right)}}{2} \]derivative simplifyDifferentiate x with respect to x. Multiply the constants (3/4 * 2 = 3/2).✓ Proved
Answer \( \frac{3}{2} - \frac{3 \cos{\left(2 x + 2 \right)}}{2} \)
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. A reviewer's objection to the explanation stands; see 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 |
| 12 | ✓ 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 9 incorrectly drops the constant term and applies the constant rule to 2*x+2. The derivative of 2*x+2 is 2, not the derivative of 2. This step applies more than one rule at once and mislabels the operation.qwen3.6:27b-mlx: fail (error) — Step 9 is mathematically incorrect; it claims to differentiate the constant 2, but the derivative of a constant is 0, not 1. The step incorrectly transforms Derivative(2*x + 2, x) into Derivative(2*x, x) by dropping the constant term without accounting for its zero derivative, and the label 'constant' is misapplied to this invalid operation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 9 is mathematically incorrect; it claims to differentiate the constant 2, but the derivative of a constant is 0, not 1. The step incorrectly transforms Derivative(2*x + 2, x) into Derivative(2*x, x) by dropping the constant term without accounting for its zero derivative, and the label 'constant' is misapplied to this invalid operation.gpt-oss:20b: fail (error) 2026-09-27 — Step 9 incorrectly drops the constant term and applies the constant rule to 2*x+2. The derivative of 2*x+2 is 2, not the derivative of 2. This step applies more than one rule at once and mislabels the operation.qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 9 is mathematically incorrect. It claims to apply the 'constant' rule to differentiate the constant 2, but the derivative of a constant is 0, not 1. The step incorrectly transforms Derivative(2*x + 2, x) into Derivative(2*x, x) by dropping the +2 without accounting for its derivative being 0, or it implies the derivative of the constant term is 1, which is false.gpt-oss:20b: fail (error) 2026-09-27 — Step 9 incorrectly applies the derivative: it claims to differentiate the constant 2, but the derivative of 2*x+2 is 2 (from 2*x) plus 0 from the constant. The step also uses an incorrect rule label (“constant”).
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.