Home›Calculus 1›Chain rule›Problem 2.110

Derivative of \( \displaystyle \frac{\sin{\left(3 x \right)}}{3} \)

Problem 2.110 · medium

Differentiate \( \displaystyle f(x) = \frac{\sin{\left(3 x \right)}}{3} \).

\[ \frac{d}{d x} \frac{\sin{\left(3 x \right)}}{3} \]
Start with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \sin{\left(3 x \right)}}{3} \]
constant-multiplePull out the constant factor 1/3.✓ Proved
\[ = \frac{\cos{\left(3 x \right)} \frac{d}{d x} 3 x}{3} \]
chainApply the chain rule to the sine function.✓ Proved
\[ = \cos{\left(3 x \right)} \]
derivative algebraDifferentiate the inner function 3*x. Simplify the expression by multiplying 1/3 and 3.✓ Proved

Answer \( \cos{\left(3 x \right)} \)

✓ 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
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
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the constant-multiple rule, chain rule, and basic derivative rules in separate steps. The labels are accurate and the logic is sound.

Every verdict on record (15)

qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant-multiple rule, chain rule, and basic derivative rules in separate steps. The labels are accurate and the logic is sound.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies the constant multiple rule, chain rule, and basic derivative rules in separate steps. The labels are accurate and the logic is sound.
deepseek-r1:70b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-17
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.