Derivative of \( \displaystyle - \frac{x}{2} + \frac{\sin{\left(4 x + 4 \right)}}{8} \)
Problem 2.552 · hard Video
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 \( \frac{\cos{\left(4 x + 4 \right)} - 1}{2} \)
✓ 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 |
| 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 (style) — Step 5 incorrectly labels the differentiation of 4*x as a "constant" rule; it should be a product or constant‑multiple rule. The chain rule step in line 4 is fine, but the subsequent labeling is inconsistent with the allowed vocabulary.qwen3.6:27b-mlx: fail (error) — Step 5 is labeled 'constant' but performs differentiation of the term 4x (specifically, it appears to be an incomplete or mislabeled step in the chain rule application where the derivative of the inner function is being computed). More critically, Step 5 claims to 'Differentiate the constant 4', which is mathematically nonsensical in this context; the derivative of 4 is 0, but the step transforms Derivative(4*x + 4, x) into Derivative(4*x, x), which is an algebraic simplification/rewrite, not a differentiation rule application, and the label 'constant' is incorrect for rewriting 4x+4 to 4x. Furthermore, Step 6 labels the differentiation of 4x as 'power', which is acceptable, but Step 5's logic is flawed. Actually, looking closer: Step 4 has Derivative(4*x + 4, x). Step 5 changes this to Derivative(4*x, x). This is an algebraic simplification (since derivative of constant 4 is 0, so derivative of 4x+4 is same as derivative of 4x). The label 'constant' is wrong for this algebraic step. It should be 'simplify' or 'algebra'. Also, the note says 'Differentiate the constant 4', which is false; it didn't differentiate 4, it removed it. This is a misleading note and incorrect label.
Senior review claude-sonnet-5-5, 2026-10-10: pass — Every line is SymPy-verified, the rule labels are legitimate (sum, constant, chain, constant, power, simplify, derivative), and the final line matches the stated answer. The step 5 'constant' label is appropriate because the derivative of the constant 4 is 0.
gpt-oss:20b: dismiss — Step 5 drops the additive constant 4 because its derivative is 0, so the 'constant' rule (derivative of a constant) is a legitimate label; the vocabulary does not require product or constant-multiple here.qwen3.6:27b-mlx: dismiss — Going from Derivative(4*x+4) to Derivative(4*x) applies the constant rule (derivative of 4 is 0) together with the sum rule, which is one legitimate change; the note 'Differentiate the constant 4' accurately describes it, and SymPy proved the line.(a panel model has both passed and objected to this solution under the same prompt): dismiss — This is not a specific defect, and every line is proved by SymPy with a defensible rule label.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-10 — Every line is SymPy-verified, the rule labels are legitimate (sum, constant, chain, constant, power, simplify, derivative), and the final line matches the stated answer. The step 5 'constant' label is appropriate because the derivative of the constant 4 is 0.qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 5 is labeled 'constant' but performs differentiation of the term 4x (specifically, it appears to be an incomplete or mislabeled step in the chain rule application where the derivative of the inner function is being computed). More critically, Step 5 claims to 'Differentiate the constant 4', which is mathematically nonsensical in this context; the derivative of 4 is 0, but the step transforms Derivative(4*x + 4, x) into Derivative(4*x, x), which is an algebraic simplification/rewrite, not a differentiation rule application, and the label 'constant' is incorrect for rewriting 4x+4 to 4x. Furthermore, Step 6 labels the differentiation of 4x as 'power', which is acceptable, but Step 5's logic is flawed. Actually, looking closer: Step 4 has Derivative(4*x + 4, x). Step 5 changes this to Derivative(4*x, x). This is an algebraic simplification (since derivative of constant 4 is 0, so derivative of 4x+4 is same as derivative of 4x). The label 'constant' is wrong for this algebraic step. It should be 'simplify' or 'algebra'. Also, the note says 'Differentiate the constant 4', which is false; it didn't differentiate 4, it removed it. This is a misleading note and incorrect label.gpt-oss:20b: fail (style) 2026-09-26 — Step 5 incorrectly labels the differentiation of 4*x as a "constant" rule; it should be a product or constant‑multiple rule. The chain rule step in line 4 is fine, but the subsequent labeling is inconsistent with the allowed vocabulary.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.