Home›Calculus 1›Chain rule›Problem 2.541

Derivative of \( \displaystyle \frac{\operatorname{asin}{\left(4 x \right)}}{2} \)

Problem 2.541 · medium

Differentiate \( \displaystyle f(x) = \frac{\operatorname{asin}{\left(4 x \right)}}{2} \).

\[ \frac{d}{d x} \frac{\operatorname{asin}{\left(4 x \right)}}{2} \]
constant-multipleStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \operatorname{asin}{\left(4 x \right)}}{2} \]
chainApply the chain rule to the inner function 4*x.✓ Proved
\[ = \frac{\frac{d}{d x} 4 x}{2 \sqrt{1 - 16 x^{2}}} \]
derivativeDifferentiate the outer function asin(u).✓ Proved
\[ = \frac{2}{\sqrt{1 - 16 x^{2}}} \]
derivative simplifyDifferentiate the inner function 4*x. Simplify the resulting expression.✓ Proved

Answer \( \frac{2}{\sqrt{1 - 16 x^{2}}} \)

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. A reviewer's objection to the explanation stands; see the receipt. 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 - 16x*2 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - 16x*2 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - 16x*2 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where 1 - 16x*2 = 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated f directly and got the stated answer

Reviewers

gpt-oss:20b: fail (style) — Step 2 incorrectly labels the constant‑multiple step as "chain"; step 3 incorrectly labels the application of the derivative of asin (which already includes the chain rule) as "derivative". Each step should use the correct single rule name from the allowed vocabulary.
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-10: fail — The math is correct and SymPy proved every line. The label and note on line 2 describe the wrong rule: the step is constant-multiple, while the chain rule only appears in the next step, so a student would be misled about where it is applied.

gpt-oss:20b: uphold — The step into line 2 only pulls the constant 1/2 out of the derivative, which is constant-multiple, yet it is labeled 'chain' with a note about the inner function 4*x that the line does not act on; the objection to the 'derivative' label on line 3 is dismissed, since differentiating the outer asin there is a legitimate use of that rule.

Every verdict on record (7)

claude-sonnet-5-5: fail (misleading) 2026-10-10 — The math is correct and SymPy proved every line. The label and note on line 2 describe the wrong rule: the step is constant-multiple, while the chain rule only appears in the next step, so a student would be misled about where it is applied.
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: fail (style) 2026-09-26 — Step 2 incorrectly labels the constant‑multiple step as "chain"; step 3 incorrectly labels the application of the derivative of asin (which already includes the chain rule) as "derivative". Each step should use the correct single rule name from the allowed vocabulary.
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 1 is labeled 'constant-multiple' but performs no transformation; it merely restates the problem, violating the rule that each step must change one thing. Step 2 is labeled 'chain' but only factors out the constant 1/2, which is a 'constant-multiple' step, not a chain rule application.
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: fail (error) 2026-09-21 — Step 2 incorrectly labels a constant‑multiple application as a chain rule. Step 3 applies two rules at once (derivative of asin and the chain rule), violating the one‑rule‑per‑step rule.

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.