#### Explain solution RD Sharma class 12 Chapter 11 Higher Order Derivatives Exercise Multiple Choice Question question 16

(a)

Hint:

We must have known about the derivative of trigonometric function like  $\sin^{-1}x$ .

Given:

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

Explanation:

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

Differentiate with respect to $x$

$\frac{d y}{d x}=2 \sin ^{-1} x \times \frac{1}{\sqrt{1-x^{2}}}$

Again differentiate with respect to $x$

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

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

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

$\begin{gathered} \left(1-x^{2}\right) y_{2}=2+x y_{1} \end{gathered}$