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Q.
If $x=exp\left\{tan^{-1}\left(\frac{y-x^{2}}{x^{2}}\right)\right\}$, then $\frac{dy}{dx}$ equals
Continuity and Differentiability
Solution:
Given that, $x=exp\left\{tan^{-1}\left(\frac{y-x^{2}}{x^{2}}\right)\right\}$
Taking log on both sides, we get
$log\,x=tan^{-1}\left(\frac{y-x^{2}}{x^{2}}\right)$
$\Rightarrow \frac{y-x^{2}}{x^{2}}=tan\left(log\,x\right)$
$\Rightarrow y=x^{2}\,tan\left(logx\right)+x^{2}$
Differentiating $w$.$r$.$t$. $x$, we get
$\frac{dy}{dx}=2x\,tan\left(log\,x\right)+x^{2} \frac{sec^{2}\left(log\,x\right)}{x}+2x$
$\Rightarrow \frac{dy}{dx}=2x\left[1+tan\left(log\,x\right)\right]+x\,sec^{2}\left(log\,x\right)$