Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(4 x + 2 \right)}}{8} \)
Problem 2.467 · hard
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(4 x + 2 \right)}}{8} \).
- \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(4 x + 2 \right)}}{8}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(4 x + 2 \right)}}{8} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(4 x + 2 \right)}}{8} \]constantPull out the constant factor 1/8.✓ Proved
- \[ = - \frac{\cos{\left(4 x + 2 \right)} \frac{d}{d x} \left(4 x + 2\right)}{8} + \frac{d}{d x} \frac{x}{2} \]trigApply the derivative rule for sine.✓ Proved
- \[ = - \frac{\cos{\left(4 x + 2 \right)} \frac{d}{d x} 4 x}{8} + \frac{d}{d x} \frac{x}{2} \]chainApply the chain rule to the inner function.✓ Proved
- \[ = - \frac{\cos{\left(4 x + 2 \right)} \frac{d}{d x} x}{2} + \frac{d}{d x} \frac{x}{2} \]constant algebraPull out the constant factor 4. Simplify the fraction 4/8.✓ Proved
- \[ = - \frac{\cos{\left(4 x + 2 \right)}}{2} + \frac{d}{d x} \frac{x}{2} \]derivativeDifferentiate x with respect to x.✓ Proved
- \[ = \frac{1}{2} - \frac{\cos{\left(4 x + 2 \right)}}{2} \]simplifySimplify the final expression.✓ Proved
Answer \( \frac{1}{2} - \frac{\cos{\left(4 x + 2 \right)}}{2} \)
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. 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: fail (misleading) — Step 5 incorrectly labels the operation as a chain rule. The step removes the constant 2 from the inner derivative, which is a simplification (constant‑multiple/derivative of a constant), not a chain rule application. This misleads a student into thinking the chain rule was used here.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (misleading) 2026-09-21 — Step 5 incorrectly labels the operation as a chain rule. The step removes the constant 2 from the inner derivative, which is a simplification (constant‑multiple/derivative of a constant), not a chain rule application. This misleads a student into thinking the chain rule was used here.qwen3.6:27b-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 function (4x+2) as 'chain'; the chain rule was already applied in Step 4, and Step 5 is simply expanding the derivative of the inner function, which should be labeled 'sum' or 'algebra' depending on interpretation, but 'chain' is definitely wrong here as the chain rule structure was already resolved.gpt-oss:20b: fail (error) 2026-09-20 — Step 5 incorrectly drops the constant term in the inner derivative; it changes Derivative(4*x + 2, x) to Derivative(4*x, x), which is not a valid application of any rule and results in an incorrect derivative.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 5 is labeled 'chain' but performs an algebraic simplification of the inner function (4*x + 2 -> 4*x) rather than applying the chain rule formula. The chain rule application actually occurred in Step 4, which was incorrectly labeled 'trig'. Step 5 should be labeled 'algebra' or 'simplify'.gpt-oss:20b: fail (error) 2026-09-20 — Step 5 incorrectly applies the chain rule to the inner function 4*x+2, treating it as 4*x and ignoring the constant 2. The derivative of 4*x+2 is 4, not 4*Derivative(x,x). This step applies more than one rule at once and mislabels the operation.
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.