Home›Calculus 1›Chain rule›Problem 2.892

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

Problem 2.892 · medium

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

\[ \frac{d}{d x} \frac{\operatorname{asin}{\left(3 x \right)}}{3} \]
constant-multipleApply the constant multiple rule.✓ Proved
\[ = \frac{\frac{d}{d x} \operatorname{asin}{\left(3 x \right)}}{3} \]
derivativeDifferentiate the arcsine function.✓ Proved
\[ = \frac{\frac{d}{d x} 3 x}{3 \sqrt{1 - 9 x^{2}}} \]
chainApply the chain rule.✓ Proved
\[ = \frac{1}{\sqrt{1 - 9 x^{2}}} \]
derivative algebra simplifyDifferentiate the inner function 3*x. Simplify the product in the numerator. Cancel the common factor of 3.✓ Proved

Answer \( \frac{1}{\sqrt{1 - 9 x^{2}}} \)

✓ 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 asin is real only on [-1, 1]
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 1 - 9x*2 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - 9x*2 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - 9x*2 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - 9x*2 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where 1 - 9x*2 = 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: fail — The math is correct and every line is proved. The labels for lines 1 and 2 look shifted: the step from line 1 to line 2 pulls out the constant 1/3, which is constant-multiple, but line 2 is labeled 'derivative' with the note 'Differentiate the arcsine function'. Line 1 is only the problem statement yet carries the constant-multiple label.

unnamed panel model (pass and objection under the same prompt): dismiss — No specific defect was stated, and SymPy proved every line equal to the previous one and the final line equal to the stated answer, so there is no mathematical error to uphold.

Every verdict on record (7)

claude-sonnet-5-5: fail (style) 2026-10-13 — The math is correct and every line is proved. The labels for lines 1 and 2 look shifted: the step from line 1 to line 2 pulls out the constant 1/3, which is constant-multiple, but line 2 is labeled 'derivative' with the note 'Differentiate the arcsine function'. Line 1 is only the problem statement yet carries the constant-multiple label.
qwen3.6:27b-mlx: pass 2026-09-27
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 derivative rules in separate steps. The algebraic simplifications are also handled in distinct steps as required.
gpt-oss:20b: fail (error) 2026-09-26 — Step 6 attempts to cancel a factor of 3 that no longer exists; the expression after step 5 is already 1/sqrt(1-(3*x)**2). The cancellation step is unnecessary and incorrectly labeled as "simplify".
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.