Question

Find dy/dx ,if x^y=y^x​

Answer 1

Answer:

${x}^{y} = {y}^{x} \\ taking \: log \: both \: sides \\ log( {x}^{y} ) = log( {y}^{x} ) \\ = > y log(x) = x log(y) \\ = > differentiating \: both \: sides \\ log(x) \times \frac{dy}{dx} + \frac{y}{x} = log(y) + \frac{x}{y} \times \frac{dy}{dx} \\ = > \frac{dy}{dx} ( log(x) - \frac{x}{y} ) = log(y) - \frac{y}{x} \\ = > \frac{dy}{dx} = \frac{ log(y) - \frac{y}{x} }{ log(x) - \frac{x}{y} }$