Home›Calculus 1›Chain rule›Problem 2.865

Derivative of \( \displaystyle 5 \operatorname{asin}{\left(x - 1 \right)} \)

Problem 2.865 · medium Mental math Video

Differentiate \( \displaystyle f(x) = 5 \operatorname{asin}{\left(x - 1 \right)} \).

\[ \frac{d}{d x} 5 \operatorname{asin}{\left(x - 1 \right)} \]
Start with the derivative of the function.✓ Proved
\[ = 5 \frac{d}{d x} \operatorname{asin}{\left(x - 1 \right)} \]
constant-multiplePull the constant out of the derivative.✓ Proved
\[ = \frac{5}{\sqrt{1 - \left(x - 1\right)^{2}}} \]
inverse-trig simplifyApply the derivative rule for arcsine. Simplify the expression.✓ Proved

Answer \( \frac{5}{\sqrt{x \left(2 - x\right)}} \)

Top: the graph of f = 5asin(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5/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]
3	✓ 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
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - (x - 1)**2 = 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 (error) — Step 3 applies two operations: the inverse‑trig derivative rule and an unnecessary division by 1. It should be written as 1/sqrt(1-(x-1)**2) and labeled only with "inverse-trig".
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-13: pass — Every line is correct and SymPy proved each equality. The final answer follows because 1-(x-1)^2 = 2x - x^2 = x(2-x). The only blemish is the cosmetic '/(1)' in line 3, which is not a defect.

gpt-oss:20b: dismiss — The '/(1)' is the inner derivative d/dx(x-1)=1 from the chain rule, evaluated as part of the single arcsine-derivative step, so this is not a second operation and the label 'inverse-trig' is acceptable.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-13 — Every line is correct and SymPy proved each equality. The final answer follows because 1-(x-1)^2 = 2x - x^2 = x(2-x). The only blemish is the cosmetic '/(1)' in line 3, which is not a defect.
qwen3.6:27b-mlx: pass 2026-09-27
gpt-oss:20b: fail (error) 2026-09-27 — Step 3 applies two operations: the inverse‑trig derivative rule and an unnecessary division by 1. It should be written as 1/sqrt(1-(x-1)**2) and labeled only with "inverse-trig".
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the constant multiple rule and the chain rule for the inverse sine function, followed by algebraic simplification to match the stated answer.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-26 — The steps correctly apply the constant multiple rule and the chain rule for the inverse sine function, followed by algebraic simplification to match the stated answer.
gpt-oss:20b: fail (style) 2026-09-26 — Step 3 applies two rules at once: it uses the inverse‑trig derivative and also simplifies by dividing by 1. Each step should change only one thing and label the rule applied.

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.