==>$y = \sqrt {\frac{{(1 + x)(1 + x)}}{{(1 - x)(1 + x)}}} = \sqrt {\frac{{{{(1 + x)}^2}}}{{1 - {x^2}}}} $
Differentiating with respect to $x,$ we get
$\frac{{dy}}{{dx}} = \frac{{{{(1 - x)}^{1/2}}\frac{d}{{dx}}{{(1 + x)}^{1/2}} - {{(1 + x)}^{1/2}}\frac{d}{{dx}}{{(1 - x)}^{1/2}}}}{{(1 - x)}}$
$ = \frac{{(1 - x) + (1 + x)}}{{2{{(1 - x)}^{3/2}}{{(1 + x)}^{1/2}}}}$
$x = \frac{2}{3}y$.
Trick : $\log y = \frac{1}{2}\log (1 + x) - \frac{1}{2}\log (1 - x)$
==> $\frac{1}{y}\frac{{dy}}{{dx}} = \frac{1}{{2(1 + x)}} + \frac{1}{{2(1 - x)}}$
==> $\frac{{dy}}{{dx}} = \frac{1}{{(1 + x)(1 - x)}} \times \sqrt {\frac{{1 + x}}{{1 - x}}} $
$ = \frac{1}{{{{(1 + x)}^{1/2}}{{(1 - x)}^{3/2}}}}$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.