Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(6 x - 6 \right)}}{12} \)
Problem 2.270 · hard
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(6 x - 6 \right)}}{12} \).
- \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(6 x - 6 \right)}}{12}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(6 x - 6 \right)}}{12} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(6 x - 6 \right)}}{12} \]constantPull out the constant factor.✓ Proved
- \[ = - \frac{\cos{\left(6 x - 6 \right)} \frac{d}{d x} \left(6 x - 6\right)}{12} + \frac{d}{d x} \frac{x}{2} \]chainApply the chain rule to the sine function.✓ Proved
- \[ = - \frac{\left(- \frac{d}{d x} 6 + \frac{d}{d x} 6 x\right) \cos{\left(6 x - 6 \right)}}{12} + \frac{d}{d x} \frac{x}{2} \]sumApply the difference rule inside the derivative.✓ Proved
- \[ = - \frac{\cos{\left(6 x - 6 \right)} \frac{d}{d x} x}{2} + \frac{d}{d x} \frac{x}{2} \]constant algebra algebra simplifyDifferentiate the terms inside the parentheses. Simplify the expression inside the parentheses. Multiply the constants. Simplify the fraction 6/12.✓ Proved
- \[ = - \frac{\cos{\left(6 x - 6 \right)}}{2} + \frac{d}{d x} \frac{x}{2} \]derivativeEvaluate the derivative of x.✓ Proved
- \[ = \frac{1}{2} - \frac{\cos{\left(6 x - 6 \right)}}{2} \]simplifyFinal simplification.✓ Proved
Answer \( \frac{1}{2} - \frac{\cos{\left(6 x - 6 \right)}}{2} \)
✓ 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 |
| 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 |
| 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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: fail (style) — Step 3 is labeled 'constant' but applies the constant-multiple rule (pulling 1/12 out of the derivative); the label 'constant' typically refers to the derivative of a constant being zero. Step 6 is labeled 'constant' but applies the constant-multiple rule to pull 6 out of Derivative(6*x, x); it should be labeled 'constant-multiple'.
Every verdict on record (12)
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 3 is labeled 'constant' but applies the constant-multiple rule (pulling 1/12 out of the derivative); the label 'constant' typically refers to the derivative of a constant being zero. Step 6 is labeled 'constant' but applies the constant-multiple rule to pull 6 out of Derivative(6*x, x); it should be labeled 'constant-multiple'.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 3 is labeled 'constant' but performs the constant-multiple rule; the vocabulary requires 'constant-multiple' for pulling out a constant factor. Step 6 is labeled 'constant' but applies the derivative rule to x and the constant rule to 6, violating the one-rule-per-step constraint.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 3 is labeled 'constant' but pulls out a constant factor, which should be labeled 'constant-multiple'. Step 6 is labeled 'constant' but performs differentiation of terms, which should be labeled 'derivative' or 'sum' depending on the specific operation, not 'constant'.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 3 is labeled 'constant' but pulls out a constant factor from a derivative, which should be labeled 'constant-multiple'. Step 6 is labeled 'constant' but applies differentiation rules to terms inside the derivative, which is not a valid label for that operation (it mixes differentiation and simplification).deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: fail 2026-09-17 — Step 5 labels the rule as "sum" but the note describes applying the difference rule to differentiate 6*x - 6. The rule label does not match what was actually performed.deepseek-r1:70b: pass 2026-09-17
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.