Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.610

Derivative of \( \displaystyle 2 \sin{\left(x + 1 \right)} \)

Problem 2.610 · easy Video

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 \( 2 \cos{\left(x + 1 \right)} \)

Top: the graph of f = 2sin(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2cos(x + 1).

✓ 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
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 both the trigonometric derivative rule and the chain rule in one move, but it is labeled only as "trig". Each step must change only one rule, so this is a defect.
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-10: pass — All steps are correct and SymPy proved each one. Labels are acceptable: constant-multiple, trig (or chain), derivative, simplify. The final line matches the stated answer 2*cos(x + 1).

gpt-oss:20b: dismiss — Differentiating sin(u) to cos(u)*u' is a single application of the sine rule in chain form, and the contract says the label 'trig' is acceptable here because chain-rule evaluation of a simple inner function is part of the step; the step also leaves Derivative(x+1, x) unevaluated, so it changes only one thing.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-10 — All steps are correct and SymPy proved each one. Labels are acceptable: constant-multiple, trig (or chain), derivative, simplify. The final line matches the stated answer 2*cos(x + 1).
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: fail (error) 2026-09-26 — Step 3 applies both the trigonometric derivative rule and the chain rule in one move, but it is labeled only as "trig". Each step must change only one rule, so this is a defect.
qwen3.6:27b-mlx: pass 2026-09-21
gpt-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 2026-09-21
gpt-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.