Differentiate the following functions with respect to x :

Solution:

$y=\tan ^{-1}\left\{\frac{x}{a+\sqrt{a^{2}-x^{2}}}\right\}$

Let $x=a \sin \theta$

Now

$y=\tan ^{-1}\left\{\frac{a \sin \theta}{a+\sqrt{a^{2}-a^{2} \sin ^{2} \theta}}\right\}$

Using $\sin ^{2} \theta+\cos ^{2} \theta=1$

$y=\tan ^{-1}\left\{\frac{a \sin \theta}{a+a \sqrt{\cos ^{2} \theta}}\right\}$

$y=\tan ^{-1}\left\{\frac{\sin \theta}{1+\cos \theta}\right\}$

Using $2 \cos ^{2} \theta=1+\cos \theta$ and $2 \sin \theta \cos \theta=\sin 2 \theta$

$y=\tan ^{-1}\left\{\frac{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}{2 \cos ^{2} \frac{\theta}{2}}\right\}$

$y=\tan ^{-1}\left\{\tan \frac{\theta}{2}\right\}$

Considering the limits,

$-a

$-1<\sin \theta<1$

$-\frac{\pi}{2}<\theta<\frac{\pi}{2}$

$-\frac{\pi}{4}<\frac{\theta}{2}<\frac{\pi}{4}$

Now,

$y=\tan ^{-1}\left\{\tan \frac{\theta}{2}\right\}$

$y=\frac{\theta}{2}$

$y=\frac{1}{2} \sin ^{-1} \frac{x}{a}$

Differentiating w.r.t $\mathrm{x}$, we get

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

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

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