Question

$\frac{d}{dx}\left\{{\sin }^{-1}\left[2ax\sqrt{(1–a^2{x}^2)}\right]\right\}=$

• A. $\frac{2a}{\sqrt{(a^2–x^2)}}$
• B. $\frac{a}{\sqrt{(a^2–x^2)}}$
• C. $\frac{2a}{\sqrt{(1–a^2{x}^2)}}$
• D. $\frac{a}{\sqrt{(1–a^2{x}^2)}}$

Answer 1

The correct option is C

$\frac{2a}{\sqrt{(1–a^2{x}^2)}}$

Find the differentiation of given function:

Given, $y={\sin }^{-1}\left[2ax\sqrt{(1–a^2{x}^2)}\right]$

Put $ax=\sin \theta$

$\Rightarrow$$y={\sin }^{-1}\left[2\sin \theta \sqrt{(1–{\sin }^2\theta )}\right]$

$\begin{array}{rcl}& =& {\sin }^{-1}(2\sin \theta \sqrt{{\cos }^2\theta })\\ & =& {\sin }^{-1}(2\sin \theta \cos \theta )\\ & =& {\sin }^{-1}\sin 2\theta \\ & =& 2\theta \\ & =& 2{\sin }^{-1}\left(ax\right)\end{array}$

Differentiate it with respect to $x$, we get

$\begin{array}{rcl}\therefore \frac{dy}{dx}& =& \left[\frac{2}{\sqrt{(1–a^2{x}^2})}\right]\times a\\ & =& \frac{\mathbf{2}\mathbf{a}}{\sqrt{\mathbf{(}\mathbf{1}\mathbf{}\mathbf{–}\mathbf{}{\mathbf{a}}^{\mathbf{2}}{\mathbf{x}}^{\mathbf{2}}\mathbf{)}}}\end{array}$ $\left[\mathbf{\because }\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{-}\mathbf{1}}\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{1}}{\sqrt{\mathbf{1}\mathbf{}\mathbf{-}\mathbf{}{\mathbf{x}}^{\mathbf{2}}}}\right]$

Hence, option (C) is correct.