The function defined by f(x) = sin⁻¹((1-x²)/(1+x²)) is differentiable at x = 0. Is this statement true or false? Give reason for your answer.

The given function is not differentiable at x = 0. Hence the statement is false. f(x) = sin⁻¹((1-x²)/(1+x²)) = π/2 - cos⁻¹((1-x²)/(1+x²)) f(x) = π/2 - 2tan⁻¹x for x ≥ 0 and π/2 + 2tan⁻¹x for x 0, not derivable for x = 0, 2/(1+x²) for x < 0

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Mathematics20252 marksShort answerContinuity, Differentiability and Differentiation

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The function defined by f(x) = sin⁻¹((1-x²)/(1+x²)) is differentiable at x = 0. Is this statement…

The function defined by $f(x) = \sin^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ is differentiable at $x = 0$. Is this statement true or false? Give reason for your answer.

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The given function is not differentiable at $x = 0$. Hence the statement is false. $f(x) = \sin^{-1}\left(\frac{1-x^2}{1+x^2}\right) = \frac{\pi}{2} - \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ $f(x) = \frac{\pi}{2} - 2\tan^{-1}x$ for $x \geq 0$ and $\frac{\pi}{2} + 2\tan^{-1}x$ for $x < 0$ $f'(x) = \frac{-2}{1+x^2}$ for $x > 0$, not derivable for $x = 0$, $\frac{2}{1+x^2}$ for $x < 0$