Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.905

Derivative of \( \displaystyle 5 \sin{\left(x + 2 \right)} \)

Problem 2.905 · easy Video

Differentiate \( \displaystyle f(x) = 5 \sin{\left(x + 2 \right)} \).

\[ \frac{d}{d x} 5 \sin{\left(x + 2 \right)} \]
Start with the derivative of the function.✓ Proved
\[ = 5 \frac{d}{d x} \sin{\left(x + 2 \right)} \]
constant-multiplePull the constant out of the derivative.✓ Proved
\[ = 5 \cos{\left(x + 2 \right)} \frac{d}{d x} \left(x + 2\right) \]
trigApply the derivative rule for sine.✓ Proved
\[ = 5 \cos{\left(x + 2 \right)} \]
derivative simplifyDifferentiate the inner function x + 2. Simplify the final expression.✓ Proved

Answer \( 5 \cos{\left(x + 2 \right)} \)

Top: the graph of f = 5sin(x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5cos(x + 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
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: pass
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-13: pass — Every line is correct and SymPy proved each step. The constant-multiple, trig (sine derivative with chain factor), derivative (d/dx of x+2 = 1) and simplify (multiply by 1) labels all fit what each step does. The panel objection was unspecific, so there was nothing to rule on.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-13 — Every line is correct and SymPy proved each step. The constant-multiple, trig (sine derivative with chain factor), derivative (d/dx of x+2 = 1) and simplify (multiply by 1) labels all fit what each step does. The panel objection was unspecific, so there was nothing to rule on.
qwen3.6:27b-mlx: pass 2026-09-27
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: fail (misleading) 2026-09-26 — Step 3 incorrectly labels the application of the chain rule as "trig" and conflates two rules; it should be labeled "chain" (or "derivative" with the inner function) and not combine the trig rule with the chain rule in a single step.
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: pass 2026-09-26

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 2026-09-27 with SymPy 1.14.0.