Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(4 x + 2 \right)}}{8} \)
Problem 2.467 · hard Video
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 \( \frac{1}{2} - \frac{\cos{\left(4 x + 2 \right)}}{2} \)
✓ 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. How we verify
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
Senior review claude-sonnet-5-5, 2026-10-09: pass — Every line is algebraically equivalent to the one before, and the final line matches the stated answer 1/2 - cos(4x+2)/2. The step 5 'chain' label is arguable but acceptable under the contract, since it evaluates the inner derivative as part of the chain rule.
gpt-oss:20b: dismiss — Step 5 continues the chain-rule treatment of the inner function 4x+2, and the contract counts evaluating a simple inner derivative as part of a chain-rule step; SymPy confirms the line is equal to the previous one, so there is no false statement.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-09 — Every line is algebraically equivalent to the one before, and the final line matches the stated answer 1/2 - cos(4x+2)/2. The step 5 'chain' label is arguable but acceptable under the contract, since it evaluates the inner derivative as part of the chain rule.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.