Derivative of \( \displaystyle 5 \operatorname{asin}{\left(x - 1 \right)} \)
Problem 2.865 · medium Mental math
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 \( \frac{5}{\sqrt{x \left(2 - x\right)}} \)
✓ 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 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: passqwen3.6:27b-mlx: pass — 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.
Every verdict on record (4)
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 2026-09-26qwen3.6:27b-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 2026-09-26 with SymPy 1.14.0.