Derivative of \( \displaystyle \operatorname{asin}{\left(\frac{\sqrt{4 x^{2} - 1}}{\sqrt{1 - 4 x^{2}}} \right)} \)
Problem 2.1385 · hard Beautiful
Differentiate \( \displaystyle f(x) = \operatorname{asin}{\left(\frac{\sqrt{4 x^{2} - 1}}{\sqrt{1 - 4 x^{2}}} \right)} \).
- \[ \frac{d}{d x} \operatorname{asin}{\left(\frac{\sqrt{4 x^{2} - 1}}{\sqrt{1 - 4 x^{2}}} \right)} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \operatorname{asin}{\left(\sqrt{\frac{4 x^{2} - 1}{1 - 4 x^{2}}} \right)} \]algebraSimplify the fraction inside the square root.✓ Proved
- \[ = \frac{d}{d x} i \ln{\left(1 + \sqrt{2} \right)} \]algebra simplifyNotice that (4*x**2 - 1)/(1 - 4*x**2) simplifies to -1. Replace the square root of -1 with the imaginary unit i.✓ Proved
- \[ = \frac{d}{d x} \frac{\pi}{2} \]simplifyThe arcsine of i is a constant value.✓ Proved
- \[ = 0 \]derivativeThe derivative of a constant is zero.✓ Proved
Answer \( 0 \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 - 4*x**2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 1 - 4*x**2 = 0 log is undefined for non-positive arguments |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (misleading) — Step 5 incorrectly states that asin(I)=π/2. In fact, asin(i)=i·asinh(1)=i·ln(1+√2), so the function is not a real constant and the subsequent derivative step is invalid.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The simplification in step 3 is invalid for real x; (4x^2 - 1)/(1 - 4x^2) equals -1 only if the terms are non-zero, but the domain of the original function requires 4x^2 - 1 >= 0 and 1 - 4x^2 > 0, which is impossible for real x (except potentially at boundaries where it is undefined). Even if we consider complex numbers, arcsin(i) is not pi/2 (it is i*asinh(1)). The solution incorrectly assumes the argument is a real constant leading to pi/2.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-03 — [domain objection, downgraded to style] The simplification in step 3 is invalid for real x; (4x^2 - 1)/(1 - 4x^2) equals -1 only if the terms are non-zero, but the domain of the original function requires 4x^2 - 1 >= 0 and 1 - 4x^2 > 0, which is impossible for real x (except potentially at boundaries where it is undefined). Even if we consider complex numbers, arcsin(i) is not pi/2 (it is i*asinh(1)). The solution incorrectly assumes the argument is a real constant leading to pi/2.gpt-oss:20b: fail (misleading) 2026-10-03 — Step 5 incorrectly states that asin(I)=π/2. In fact, asin(i)=i·asinh(1)=i·ln(1+√2), so the function is not a real constant and the subsequent derivative step is invalid.qwen3.6:27b-mlx: inconclusive 2026-10-03 — reviewer returned a non-objectgpt-oss:20b: fail (error) 2026-10-03 — Step 5 incorrectly states that asin(I)=π/2; the correct value is i·asinh(1). The solution therefore applies an incorrect simplification and the derivative is not zero.
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 2026-10-03 with SymPy 1.14.0.