Derivative of \( \displaystyle \frac{3 x}{2} - \frac{\sin{\left(6 x \right)}}{4} \)
Problem 2.700 · hard
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{\sin{\left(6 x \right)}}{4} \).
- \[ \frac{d}{d x} \left(\frac{3 x}{2} - \frac{\sin{\left(6 x \right)}}{4}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{3 x}{2} - \frac{d}{d x} \frac{\sin{\left(6 x \right)}}{4} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{3 x}{2} - \frac{\frac{d}{d x} \sin{\left(6 x \right)}}{4} \]constantFactor out the constant 1/4.✓ Proved
- \[ = \frac{3 \frac{d}{d x} x}{2} - \frac{\frac{d}{d x} \sin{\left(6 x \right)}}{4} \]constant-multipleFactor out the constant 3/2.✓ Proved
- \[ = \frac{3}{2} - \frac{\frac{d}{d x} \sin{\left(6 x \right)}}{4} \]derivativeDifferentiate x.✓ Proved
- \[ = - \frac{\cos{\left(6 x \right)} \frac{d}{d x} 6 x}{4} + \frac{3}{2} \]trigApply the derivative rule for sine.✓ Proved
- \[ = \frac{3}{2} - \frac{3 \cos{\left(6 x \right)}}{2} \]derivative algebra simplifyDifferentiate 6*x. Multiply the constants. Simplify the fraction 6/4.✓ Proved
Answer \( 3 \sin^{2}{\left(3 x \right)} \)
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: passqwen3.6:27b-mlx: fail (error) — The final answer provided (3*sin(3*x)**2) does not match the result of the derivation steps (3/2 - 3/2*cos(6*x)). Although the steps are algebraically correct and lead to a valid antiderivative form via trigonometric identities, the stated answer is presented as the final result of the differentiation process without the necessary simplification step shown, and more importantly, the prompt asks to review the solution against the contract. The contract requires checking the steps. The steps themselves are valid. However, the 'Stated answer' is part of the solution package. Is the mismatch between the last step and the stated answer a defect? The prompt says 'A computer algebra system has already confirmed that consecutive lines are algebraically equal'. It does not confirm the stated answer equals the last line. But usually, the 'Stated answer' is the target. Let's look closer at the steps. Step 6 uses rule 'trig' to differentiate sin(6x). The derivative of sin(u) is cos(u)*u'. This is the chain rule. The vocabulary has 'chain'. Using 'trig' is acceptable if it implies the derivative of the trig function, but 'chain' is more precise for the composite function. However, the instructions say 'A label is correct if it names a rule the step legitimately applied.' Differentiating sin is a trig rule. Applying chain rule is a chain rule. Step 6 does both: it identifies the derivative of sin as cos AND applies the chain rule structure. This violates the 'ONE thing' rule. Step 6 applies the derivative of sine (trig) AND the chain rule structure. It should be split or labeled 'chain'. But wait, step 7 differentiates 6x. So step 6 is just replacing d/dx sin(6x) with cos(6x)*d/dx(6x). This is the Chain Rule. Labeling it 'trig' is misleading/incomplete because it hides the chain rule application which is the core complexity. But is it a 'defect'? The instructions say 'calling a product-rule step chain is a defect'. Here, calling a chain-rule step 'trig' is likely a defect because 'trig' usually refers to basic derivatives like d/dx sin(x) = cos(x). Here we have a composite. The step applies the chain rule. The label 'trig' does not name the chain rule. Therefore, the label is incorrect for the rule applied (Chain Rule). The vocabulary has 'chain'. The step applies the chain rule. The label is 'trig'. This is a labeling defect. Additionally, the final stated answer is not reached in the steps. The last step is 3/2 - 3/2*cos(6*x). The stated answer is 3*sin(3*x)**2. These are equal (using half-angle identities: 1-cos(2A) = 2sin^2(A) => 3/2(1-cos(6x)) = 3/2 * 2sin^2(3x) = 3sin^2(3x)). But the step from 9 to the stated answer is missing. The solution is incomplete. An incomplete solution is a defect. The task is to differentiate. The steps stop at an intermediate form. The stated answer is the final form. The gap is not bridged. This is an error in the solution's completeness.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — The final answer provided (3*sin(3*x)**2) does not match the result of the derivation steps (3/2 - 3/2*cos(6*x)). Although the steps are algebraically correct and lead to a valid antiderivative form via trigonometric identities, the stated answer is presented as the final result of the differentiation process without the necessary simplification step shown, and more importantly, the prompt asks to review the solution against the contract. The contract requires checking the steps. The steps themselves are valid. However, the 'Stated answer' is part of the solution package. Is the mismatch between the last step and the stated answer a defect? The prompt says 'A computer algebra system has already confirmed that consecutive lines are algebraically equal'. It does not confirm the stated answer equals the last line. But usually, the 'Stated answer' is the target. Let's look closer at the steps. Step 6 uses rule 'trig' to differentiate sin(6x). The derivative of sin(u) is cos(u)*u'. This is the chain rule. The vocabulary has 'chain'. Using 'trig' is acceptable if it implies the derivative of the trig function, but 'chain' is more precise for the composite function. However, the instructions say 'A label is correct if it names a rule the step legitimately applied.' Differentiating sin is a trig rule. Applying chain rule is a chain rule. Step 6 does both: it identifies the derivative of sin as cos AND applies the chain rule structure. This violates the 'ONE thing' rule. Step 6 applies the derivative of sine (trig) AND the chain rule structure. It should be split or labeled 'chain'. But wait, step 7 differentiates 6x. So step 6 is just replacing d/dx sin(6x) with cos(6x)*d/dx(6x). This is the Chain Rule. Labeling it 'trig' is misleading/incomplete because it hides the chain rule application which is the core complexity. But is it a 'defect'? The instructions say 'calling a product-rule step chain is a defect'. Here, calling a chain-rule step 'trig' is likely a defect because 'trig' usually refers to basic derivatives like d/dx sin(x) = cos(x). Here we have a composite. The step applies the chain rule. The label 'trig' does not name the chain rule. Therefore, the label is incorrect for the rule applied (Chain Rule). The vocabulary has 'chain'. The step applies the chain rule. The label is 'trig'. This is a labeling defect. Additionally, the final stated answer is not reached in the steps. The last step is 3/2 - 3/2*cos(6*x). The stated answer is 3*sin(3*x)**2. These are equal (using half-angle identities: 1-cos(2A) = 2sin^2(A) => 3/2(1-cos(6x)) = 3/2 * 2sin^2(3x) = 3sin^2(3x)). But the step from 9 to the stated answer is missing. The solution is incomplete. An incomplete solution is a defect. The task is to differentiate. The steps stop at an intermediate form. The stated answer is the final form. The gap is not bridged. This is an error in the solution's completeness.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (error) 2026-09-21 — The final answer is mathematically incorrect; the derivative of 3x/2 - sin(6x)/4 is 3/2 - (3/2)cos(6x), not 3sin(3x)^2. Additionally, Step 3 incorrectly labels the extraction of a constant factor as 'constant' instead of 'constant-multiple'.gpt-oss:20b: fail (error) 2026-09-21 — The final answer 3*sin(3*x)**2 does not match the derivative computed in the steps, which is 3/2 - (3/2)*cos(6*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 2026-09-26 with SymPy 1.14.0.