Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.881

Derivative of \( \displaystyle \frac{\sin{\left(4 x + 1 \right)}}{2} \)

Problem 2.881 · easy Video

Differentiate \( \displaystyle f(x) = \frac{\sin{\left(4 x + 1 \right)}}{2} \).

\[ \frac{d}{d x} \frac{\sin{\left(4 x + 1 \right)}}{2} \]
Start with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \sin{\left(4 x + 1 \right)}}{2} \]
constant-multiplePull out the constant factor 1/2.✓ Proved
\[ = \frac{\cos{\left(4 x + 1 \right)} \frac{d}{d x} \left(4 x + 1\right)}{2} \]
chainApply the chain rule.✓ Proved
\[ = 2 \cos{\left(4 x + 1 \right)} \]
derivative algebraDifferentiate the inner function 4*x + 1. Simplify the coefficients.✓ Proved

Answer \( 2 \cos{\left(4 x + 1 \right)} \)

Top: the graph of f = sin(4x + 1)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2cos(4x + 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the constant multiple rule, chain rule, and basic derivative rules in separate steps. The final simplification is algebraically correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the constant multiple rule, chain rule, and basic derivative rules in separate steps. The final simplification is algebraically correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the constant multiple rule, chain rule, and basic derivative rules in separate steps. The labels are accurate and the algebraic simplification is correct.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the constant multiple rule, chain rule, and basic derivative rules in separate steps. The algebraic simplification is correct.
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.