Question

Explain this .. how it's ans. will be dy/dx = y²/ X(1 - y ln x)​

Answer 1

Answer:

Let

$y = x ^{y}$

$applying \: log \: on \: both \: sides$

$log(y) = log( {x}^{y} ) = y log(x)$

Therefore

$\frac{1}{y} \frac{dy}{dx} = \frac{dy}{dx} log(x) + y \frac{1}{x} \\ = \frac{dy}{dx} log(x) + \frac{y}{x}$

$\frac{1}{y} \frac{dy}{dx} - \frac{dy}{dx} log(x) = \frac{y}{x}$

$\frac{dy}{dx}( \frac{1}{y} - log(x) ) = \frac{y}{x}$

Hence

$\frac{dy}{dx} = \frac{y}{x( \frac{1}{y} - log(x) ) }$

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