Write the correct alternative in the following:

Question:

Write the correct alternative in the following:

If $f(x)=\frac{\sin ^{-1} x}{\sqrt{\left(1-x^{2}\right)}}$ then $\left(1-x^{2}\right) f^{\prime}(x)-x f(x)=$

A. 1

B. $-1$

C. 0

D. none of these

Solution:

Given:

$y=f(x)=\frac{\sin ^{-1} x}{\sqrt{\left(1-x^{2}\right)}}$

$\frac{d y}{d x}=\frac{1}{\left(\sqrt{\left.\left(1-x^{2}\right)\right)^{2}}\right.}\left\{\frac{1}{\sqrt{\left(1-x^{2}\right)}} \sqrt{\left(1-x^{2}\right)}-\sin ^{-1} x \frac{(-2 x)}{2 \sqrt{\left(1-x^{2}\right)}}\right\}$

$=\frac{1}{\left(\sqrt{\left.\left(1-x^{2}\right)\right)^{2}}\right.}\left\{1+\frac{x \sin ^{-1} x}{\sqrt{\left(1-x^{2}\right)}}\right\}$

$=\frac{1+x y}{\left(1-x^{2}\right)}$

$f^{\prime}(x)=\frac{1+x f(x)}{\left(1-x^{2}\right)}$

$\left(1-x^{2}\right) f^{\prime}(x)=1+x f(x)$

$\left(1-x^{2}\right) f^{\prime}(x)-x f(x)=1$

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