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Q.
If $x^x=y^y$, then $\frac{dy}{dx}$ is equal to
Continuity and Differentiability
Solution:
Given $x^x=y^y$, taking log on the both sides, we get
$logx^x = logy^y$
$ \Rightarrow xlogx = ylogy$,
Differentiating $w$.$r$.$t$. $x$, we get
$x\left(\frac{1}{x}\right)+log\,x\cdot1=y\left(\frac{1}{y} \frac{dy}{dx}\right)+\left(log\,y\right) \frac{dy}{dx}$
$\Rightarrow 1+logx=\left(1+logy\right) \frac{dy}{dx}$
$\Rightarrow \frac{dy}{dx}=\frac{1+log\,x}{1+log\,y}$.