# Differentiate the following functions with respect to x :

Question:

Differentiate the following functions with respect to $x$ :

$\tan ^{-1}\left\{\frac{\mathrm{x}}{\sqrt{\mathrm{a}^{2}-\mathrm{x}^{2}}}\right\}, \mathrm{a}<\mathrm{x}<\mathrm{a}$

Solution:

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

Let $x=a \sin \theta$

Now

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

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

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

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

$y=\tan ^{-1}(\tan \theta)$

Considering the limits,

$-a$-a

$-1<\sin \theta<1$

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

Now, $y=\tan ^{-1}(\tan \theta)$

$y=\theta$

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

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

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

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

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