Derivative of \( \displaystyle - \operatorname{asin}{\left(x \right)} \)
Problem 2.1071 · medium
Differentiate \( \displaystyle f(x) = - \operatorname{asin}{\left(x \right)} \).
- \[ \frac{d}{d x} \left(- \operatorname{asin}{\left(x \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \operatorname{asin}{\left(x \right)} \]constantPull the constant factor out of the derivative.✓ Proved
- \[ = - \frac{1}{\sqrt{1 - x^{2}}} \]derivativeApply the derivative rule for arcsine.✓ Proved
Answer \( - \frac{1}{\sqrt{1 - x^{2}}} \)
✓ 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.
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**2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 1 - x**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 application of the derivative of arcsine as a "constant" rule; it should be labeled "derivative". The step does not apply two rules at once, but the label misidentifies the rule used.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28gpt-oss:20b: fail (style) 2026-09-28 — Step 2 incorrectly labels the application of the derivative of arcsine as a "constant" rule; it should be labeled "derivative". The step does not apply two rules at once, but the label misidentifies the rule used.qwen3.6:27b-mlx: pass 2026-09-28gpt-oss:20b: pass 2026-09-28
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 2026-09-28 with SymPy 1.14.0.