Home›Calculus 1›Chain rule›Problem 2.598

Derivative of \( \displaystyle \operatorname{asin}{\left(x - 1 \right)} \)

Problem 2.598 · medium Video

Differentiate \( \displaystyle f(x) = \operatorname{asin}{\left(x - 1 \right)} \).

\[ \frac{d}{d x} \operatorname{asin}{\left(x - 1 \right)} \]
Start with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \left(x - 1\right)}{\sqrt{1 - \left(x - 1\right)^{2}}} \]
chainApply the chain rule.✓ Proved
\[ = \frac{1}{\sqrt{1 - \left(x - 1\right)^{2}}} \]
derivativeDifferentiate the inner function x - 1.✓ Proved
\[ = \frac{1}{\sqrt{- x^{2} + 2 x}} \]
algebra simplifyExpand the squared term inside the square root. Simplify the expression under the square root.✓ Proved

Answer \( \frac{1}{\sqrt{x \left(2 - x\right)}} \)

Top: the graph of f = asin(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 1/sqrt(1 - (x - 1)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] undefined where 1 - (x - 1)**2 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - (x - 1)**2 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - (x - 1)2 = 0 undefined where -x2 + 2*x = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where -x*2 + 2x = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where x*(2 - x) = 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 1 is labeled "unlabelled" but it applies the derivative rule. It should be labeled "derivative" to match the allowed vocabulary.
qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule and simplifies the algebraic expression. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the chain rule and simplifies the algebraic expression. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: fail (style) 2026-09-26 — Step 1 is labeled "unlabelled" but it applies the derivative rule. It should be labeled "derivative" to match the allowed vocabulary.
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the chain rule, differentiates the inner function, and simplifies the algebraic expression to match the stated answer. Each step adheres to the single-change constraint and uses valid labels.
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the chain rule and simplifies the algebraic expression. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-09-21

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.