Derivative of \( \displaystyle 2 \sin{\left(x + 1 \right)} \)
Problem 2.610 · easy
Differentiate \( \displaystyle f(x) = 2 \sin{\left(x + 1 \right)} \).
- \[ \frac{d}{d x} 2 \sin{\left(x + 1 \right)} \]Start with the derivative of the function.✓ Proved
- \[ = 2 \frac{d}{d x} \sin{\left(x + 1 \right)} \]constant-multiplePull the constant out of the derivative.✓ Proved
- \[ = 2 \cos{\left(x + 1 \right)} \frac{d}{d x} \left(x + 1\right) \]trigApply the derivative rule for sine.✓ Proved
- \[ = 2 \cos{\left(x + 1 \right)} \]derivative simplifyDifferentiate the inner function x + 1. Simplify the final expression.✓ Proved
Answer \( 2 \cos{\left(x + 1 \right)} \)
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 |
| 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 3 applies two rules at once: the sine‑derivative rule (trig) and the chain rule (derivative of the inner function). It should be split into a trig step producing 2*cos(x+1)*Derivative(x+1,x) and a separate derivative step for the inner function.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (error) 2026-09-21 — Step 3 applies two rules at once: the sine‑derivative rule (trig) and the chain rule (derivative of the inner function). It should be split into a trig step producing 2*cos(x+1)*Derivative(x+1,x) and a separate derivative step for the inner function.qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (style) 2026-09-21 — Step 3 applies the chain rule (derivative of sin(u) = cos(u)·u'), but it is labeled only as "trig". The correct label should be "chain". This mislabeling could mislead a student about which rule is being used.
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.