Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(4 x - 6 \right)}}{8} \)
Problem 2.469 · hard
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(4 x - 6 \right)}}{8} \).
- \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(4 x - 6 \right)}}{8}\right) \]sumStart with the derivative of the entire function.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(4 x - 6 \right)}}{8} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(4 x - 6 \right)}}{8} \]constantPull out the constant factor 1/8.✓ Proved
- \[ = - \frac{\cos{\left(4 x - 6 \right)} \frac{d}{d x} \left(4 x - 6\right)}{8} + \frac{d}{d x} \frac{x}{2} \]chainApply the chain rule to the sine term.✓ Proved
- \[ = - \frac{\cos{\left(4 x - 6 \right)} \frac{d}{d x} 4 x}{8} + \frac{d}{d x} \frac{x}{2} \]sumApply the sum rule to the inner function.✓ Proved
- \[ = - \frac{\cos{\left(4 x - 6 \right)} \frac{d}{d x} x}{2} + \frac{d}{d x} \frac{x}{2} \]constant algebra algebraPull out the constant factor 4. Simplify the constant coefficients. Simplify 4/8 to 1/2.✓ Proved
- \[ = - \frac{\cos{\left(4 x - 6 \right)}}{2} + \frac{d}{d x} \frac{x}{2} \]derivativeEvaluate the derivative of x.✓ Proved
- \[ = \frac{1}{2} - \frac{\cos{\left(4 x - 6 \right)}}{2} \]simplifyFinal simplification.✓ Proved
Answer \( \frac{1}{2} - \frac{\cos{\left(4 x - 6 \right)}}{2} \)
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 |
| 10 | ✓ 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
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 3 is labeled 'constant' but performs a constant-multiple extraction; the vocabulary requires 'constant-multiple' for this operation. Step 6 is labeled 'constant' but performs a constant-multiple extraction from the derivative of 4x, which should also be 'constant-multiple'.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 incorrectly labels the extraction of the constant factor 1/8 as 'constant'; the correct label is 'constant-multiple'. Step 5 incorrectly labels the differentiation of the inner linear term (4*x - 6) as 'sum'; it should be 'derivative' or 'sum' applied to the derivative operator, but the step actually performs the differentiation of 4x and -6, which is a mix of rules or simply 'derivative' if viewed as a whole, but primarily the label 'sum' is misleading for differentiating a linear term inside a chain rule context where the focus is on the derivative of the inner function. More critically, Step 6 labels pulling out the 4 as 'constant', but it is part of the differentiation of 4x, which should be handled via 'constant-multiple' or 'derivative'. The most glaring error is Step 3: pulling out a constant multiplier is 'constant-multiple', not 'constant' (which usually refers to the derivative of a constant being zero). Step 5 is also flawed: differentiating 4*x - 6 is not just 'sum', it involves 'derivative' of x and 'constant' for -6. However, the contract says 'sum' is acceptable for linearity. The main defect is Step 3 using 'constant' instead of 'constant-multiple'.gpt-oss:20b: pass 2026-09-20
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.