Find d y d x , if x ^ y = y ^ x - Vidyadip

Question

Find $\frac{d y}{d x}$, if $x ^{ y }= y ^{ x }$

✓

Answer

Given $x ^{ y }= y ^{ x }$
Taking logarithm of both sides, we get
$ \log x^y=\log y^x$
$\therefore y \log x=x \log y $
Differentiating both sides w.r.t.x, we get
$ \frac{d}{d x}(y \log x)=\frac{d}{d x}(x \log y)$
$\therefore y \cdot \frac{d}{d x}(\log x)+\frac{d}{d x}(y)=x \cdot \frac{d}{d x}(\log y)+\log y \cdot \frac{d}{d x}(x)$
$\therefore y \cdot \frac{1}{x}+\log x \cdot \frac{d y}{d x}=x \cdot \frac{1}{y} \cdot \frac{d y}{d x}+\log y \cdot 1$
$\therefore\left(\log x-\frac{x}{y}\right) \frac{d y}{d x}=\left(\log y-\frac{y}{x}\right) $
$\therefore\left(\frac{y \log x-x}{y}\right) \frac{d y}{d x}=\frac{x \log y-y}{x}$
$\therefore \frac{d y}{d x}=\left(\frac{x \log y-y}{x}\right) \times\left(\frac{y}{y \log x-x}\right)$
$\therefore \frac{d y}{d x}=\frac{y}{x}\left(\frac{x \log y-y}{y \log x-x}\right)$

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