Question:
Differentiate the following functions with respect to $x$ :
$\sin ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)+\sec ^{-1}\left(\frac{1+x^{2}}{1-x^{2}}\right), x \in R$
Solution:
$y=\sin ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)+\sec ^{-1}\left(\frac{1+x^{2}}{1-x^{2}}\right)$
Using, $\sec ^{-1} x=\frac{1}{\cos ^{-1} x}$
$y=\sin ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)+\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$
Using, $\cos ^{-1} x+\sin ^{-1} x=\frac{\pi}{2}$
$y=\frac{\pi}{2}$
Differentiating w.r.t $x$ we get
$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\pi}{2}\right)$
$\frac{\mathrm{dy}}{\mathrm{dx}}=0$