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,