Derivative of \( \displaystyle - \frac{x}{2} + \frac{\sin{\left(4 x + 4 \right)}}{8} \)
Problem 2.552 · hard
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(4 x + 4 \right)}}{8} \).
- \[ \frac{d}{d x} \left(- \frac{x}{2} + \frac{\sin{\left(4 x + 4 \right)}}{8}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \frac{\sin{\left(4 x + 4 \right)}}{8} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{\frac{d}{d x} \sin{\left(4 x + 4 \right)}}{8} \]constantPull out the constant factor 1/8.✓ Proved
- \[ = \frac{\cos{\left(4 x + 4 \right)} \frac{d}{d x} \left(4 x + 4\right)}{8} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]chainApply the chain rule to the sine term.✓ Proved
- \[ = \frac{\cos{\left(4 x + 4 \right)} \frac{d}{d x} 4 x}{8} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]constantDifferentiate the constant 4.✓ Proved
- \[ = \frac{\cos{\left(4 x + 4 \right)}}{2} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]power simplifyDifferentiate 4*x. Simplify the fraction 4/8.✓ Proved
- \[ = \frac{\cos{\left(4 x + 4 \right)}}{2} - \frac{1}{2} \]derivativeEvaluate the derivative of -x/2.✓ Proved
Answer \( \frac{\cos{\left(4 x + 4 \right)} - 1}{2} \)
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 |
| 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 drops the constant term 4 from the derivative of 4*x+4, applying two rules at once and producing an incorrect intermediate expression.qwen3.6:27b-mlx: fail (error) — Step 5 is mathematically incorrect; it replaces Derivative(4*x + 4, x) with Derivative(4*x, x), which ignores the derivative of the constant 4 (which is 0, but the step implies the constant term vanishes without justification or is treated as if it were not part of the expression). Furthermore, Step 3 uses the label 'constant' to pull out a factor, but the standard label for pulling out a constant multiplier is 'constant-multiple'. While 'constant-multiple' is in the vocabulary, 'constant' usually refers to the derivative of a constant being zero. However, the primary defect is the algebraic/logical error in Step 5 where the inner derivative is incorrectly simplified by dropping the constant term inside the derivative operator without applying the sum rule or evaluating the derivative of the constant separately.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 5 is mathematically incorrect; it replaces Derivative(4*x + 4, x) with Derivative(4*x, x), which ignores the derivative of the constant 4 (which is 0, but the step implies the constant term vanishes without justification or is treated as if it were not part of the expression). Furthermore, Step 3 uses the label 'constant' to pull out a factor, but the standard label for pulling out a constant multiplier is 'constant-multiple'. While 'constant-multiple' is in the vocabulary, 'constant' usually refers to the derivative of a constant being zero. However, the primary defect is the algebraic/logical error in Step 5 where the inner derivative is incorrectly simplified by dropping the constant term inside the derivative operator without applying the sum rule or evaluating the derivative of the constant separately.gpt-oss:20b: fail (error) 2026-09-21 — Step 5 incorrectly drops the constant term 4 from the derivative of 4*x+4, applying two rules at once and producing an incorrect intermediate expression.qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 5 is mathematically incorrect: it claims to differentiate the constant 4, but the expression `Derivative(4*x + 4, x)` becomes `Derivative(4*x, x)`, which is an algebraic simplification (dropping the constant term inside the derivative), not a differentiation of a constant. The label 'constant' is wrong for this operation, and the logic implies `d/dx(4) = 0` was applied inside the derivative operator incorrectly or the step is just a mislabeled algebraic simplification.gpt-oss:20b: fail (error) 2026-09-21 — Step 5 incorrectly applies the constant rule to the term 4*x; the derivative of 4*x is 4, not 0, and the rule used should be derivative (or constant‑multiple) rather than constant. Additionally, step 6 labels the operation as "power", which is inappropriate for differentiating a linear term.
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.