explain solution RD Sharma class 12 chapter Differentiation exercise 10.2 question 74 maths

Hint: you must know the rules of solving derivatives.

Given: Prove that
$\frac{d}{d x}\left\{\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}\right\}=\sqrt{a^{2}-x^{2}}$
Solution:

$\text { L.H.S } \Rightarrow \frac{d}{d x}\left\{\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}\right\}$

$\Rightarrow \frac{d}{d x}\left(\frac{x}{2} \sqrt{a^{2}-x^{2}}\right)+\frac{d}{d x}\left(\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}\right)$

$\Rightarrow \frac{1}{2}\left[x \frac{d}{d x} \sqrt{a^{2}-x^{2}}+\sqrt{a^{2}-x^{2}} \frac{d}{d x}(x)\right]+\frac{a^{2}}{2} \times \frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^{2}}}\left(\frac{1}{a}\right)$

$\Rightarrow \frac{1}{2}\left[x \times \frac{1}{2 \sqrt{a^{2}-x^{2}}} \frac{d}{d x}\left(a^{2}-x^{2}\right)+\sqrt{a^{2}-x^{2}}\right]+\frac{a^{2}}{2} \times \frac{1}{\sqrt{\frac{a^{2}-x^{2}}{a^{2}}}}\left(\frac{1}{a}\right)$

$\Rightarrow \frac{1}{2}\left[\frac{-2 x^{2}}{2 \sqrt{a^{2}-x^{2}}}+\sqrt{a^{2}-x^{2}}\right]+\left(\frac{a^{2}}{2}\right) \frac{a}{\sqrt{a^{2}-x^{2}}} \times\left(\frac{1}{a}\right)$

$\Rightarrow \frac{1}{2}\left[\frac{-2 x^{2}+2\left|a^{2}-x^{2}\right|}{2 \sqrt{a^{2}-x^{2}}}\right]+\frac{a^{2}}{2 \sqrt{a^{2}-x^{2}}}$

$\Rightarrow \frac{a^{2}-x^{2}-x^{2}}{2 \sqrt{a^{2}-x^{2}}}+\frac{a^{2}}{2 \sqrt{a^{2}-x^{2}}}$

$\Rightarrow \frac{2 a^{2}-2 x^{2}}{2 \sqrt{a^{2}-x^{2}}} \quad[\therefore x=\sqrt{x} \times \sqrt{x}]$

\begin{aligned} &\Rightarrow \frac{\left(a^{2}-x^{2}\right)}{\sqrt{a^{2}-x^{2}}} \\\\ &\Rightarrow \sqrt{a^{2}-x^{2}} \quad \Rightarrow \text { R.H.S } \end{aligned}

∴ Proved