Derivative of \( \displaystyle \operatorname{asin}{\left(\frac{\sqrt{1 - 4 x^{2}}}{\sqrt{4 x^{2} - 1}} \right)} \)
Problem 2.534 · hard Beautiful
Differentiate \( \displaystyle f(x) = \operatorname{asin}{\left(\frac{\sqrt{1 - 4 x^{2}}}{\sqrt{4 x^{2} - 1}} \right)} \).
- \[ \frac{d}{d x} \operatorname{asin}{\left(\frac{\sqrt{1 - 4 x^{2}}}{\sqrt{4 x^{2} - 1}} \right)} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \operatorname{asin}{\left(\sqrt{\frac{1 - 4 x^{2}}{4 x^{2} - 1}} \right)} \]algebra algebraCombine the square roots into one. Factor out a negative sign from the numerator.✓ Proved
- \[ = \frac{d}{d x} i \ln{\left(1 + \sqrt{2} \right)} \]simplify simplifySimplify the expression inside the square root. Replace -1 with the imaginary unit I.✓ 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 4*x**2 - 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 4*x**2 - 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 4*x**2 - 1 = 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 log is undefined for non-positive arguments |
| 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: passqwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The simplification in step 3 is invalid because the ratio (1 - 4*x**2)/(4*x**2 - 1) equals -1 only when the denominator is non-zero, but the original function involves square roots of these terms which are undefined or imaginary for real x where the terms inside are negative. More critically, step 4 simplifies sqrt(-1) to I, treating the argument as a constant complex number, but the original expression is not a constant function over its real domain (where it is undefined) or complex domain. The function f(x) is not identically equal to asin(I); the algebraic manipulation sqrt(a)/sqrt(b) = sqrt(a/b) is not valid for all complex numbers without branch cut considerations, and even if it were, the resulting constant derivative is only valid if the function is constant, which it is not on any open interval of the real line (it is undefined). The solution incorrectly treats a non-constant or undefined expression as a constant.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — [domain objection, downgraded to style] The simplification in step 3 is invalid because the ratio (1 - 4*x**2)/(4*x**2 - 1) equals -1 only when the denominator is non-zero, but the original function involves square roots of these terms which are undefined or imaginary for real x where the terms inside are negative. More critically, step 4 simplifies sqrt(-1) to I, treating the argument as a constant complex number, but the original expression is not a constant function over its real domain (where it is undefined) or complex domain. The function f(x) is not identically equal to asin(I); the algebraic manipulation sqrt(a)/sqrt(b) = sqrt(a/b) is not valid for all complex numbers without branch cut considerations, and even if it were, the resulting constant derivative is only valid if the function is constant, which it is not on any open interval of the real line (it is undefined). The solution incorrectly treats a non-constant or undefined expression as a constant.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (style) 2026-09-20 — Step 4 applies two simplifications at once (simplifying the argument and evaluating the square root of -1). It should be split into a separate simplification step that replaces sqrt(-1) with I, and then a derivative step that recognizes the constant.qwen3.6:27b-mlx: inconclusive 2026-09-20 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The simplification in step 3 is invalid because the term (4*x**2 - 1)/(4*x**2 - 1) equals 1 only when the denominator is non-zero, but the domain ofgpt-oss:20b: fail (style) 2026-09-20 — Step 1 is missing a rule label; every step must name a rule from the allowed vocabulary.
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.