Derivative of \( \displaystyle \sin{\left(2 x + 2 \right)} \)
Problem 2.249 · medium
Differentiate \( \displaystyle f(x) = \sin{\left(2 x + 2 \right)} \).
- \[ \frac{d}{d x} \sin{\left(2 x + 2 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \cos{\left(2 x + 2 \right)} \frac{d}{d x} \left(2 x + 2\right) \]chainApply the chain rule.✓ Proved
- \[ = \left(\frac{d}{d x} 2 + \frac{d}{d x} 2 x\right) \cos{\left(2 x + 2 \right)} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = \left(\frac{d}{d x} 2 + 2\right) \cos{\left(2 x + 2 \right)} \]powerDifferentiate 2*x.✓ Proved
- \[ = 2 \cos{\left(2 x + 2 \right)} \]constant simplifyThe derivative of a constant is zero. Simplify the final expression.✓ Proved
Answer \( 2 \cos{\left(2 x + 2 \right)} \)
✓ 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.
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 |
| 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 4 incorrectly labels the derivative of 2*x as a "power" rule; it should be a "constant-multiple" rule. The chain of labels should reflect single-rule applications, and the current labeling misidentifies the rule used.deepseek-r1:70b: passqwen3.6:27b-mlx: pass — The solution correctly applies the chain rule, sum rule, power rule, and constant rule in separate steps. All labels are valid and accurately describe the single transformation performed in each step.
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule, sum rule, power rule, and constant rule in separate steps. All labels are valid and accurately describe the single transformation performed in each step.gpt-oss:20b: fail (style) 2026-09-20 — Step 4 incorrectly labels the derivative of 2*x as a "power" rule; it should be a "constant-multiple" rule. The chain of labels should reflect single-rule applications, and the current labeling misidentifies the rule used.qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (style) 2026-09-20 — Step 4 applies the wrong rule label: differentiating 2*x uses the constant‑multiple rule, not the power rule. The note also incorrectly states “Differentiate 2*x” under a power rule.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the chain rule, sum rule, power rule, and constant rule in separate steps. The labels accurately reflect the operations performed.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (style) 2026-09-19 — Step 4 incorrectly labels the rule as "power"; the derivative of 2*x is obtained by the constant‑multiple rule, not a power rule. The note also incorrectly states "Differentiate 2*x" instead of applying the constant‑multiple rule.qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (style) 2026-09-19 — Step 4 incorrectly labels the derivative of 2*x with the rule "power"; the correct rule is "constant-multiple" (or "product" if treating 2*x as 2·x).gpt-oss:20b: fail 2026-09-17 — Step 4 incorrectly labels the operation as a "power" rule; differentiating 2*x is a constant‑multiple rule, not a power rule.deepseek-r1:70b: pass 2026-09-17
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.