Question

differentiate (x+1/x) to power of x with respect to x​

Answer 1

$HEYA \: \\ \\ GIVEN \: \: QUESTION \: \: Is \: \: \\ \\ y = (x + \frac{1}{x} ) {}^{x} \: \: \: find \: \: \: \frac{dy}{dx} \\ \\ taking \: log \: on \: both \: sides \: we \: have \\ \\ log(y) = log(x + \frac{1}{ x } ) {}^{x} \\ \\ log(y) = x log(x + \frac{1}{x} ) \\ Differentiate \: \: both \: sides \: with \: respect \: to \: \\ x \: we \: have \\ \\ \frac{1}{y} \frac{dy}{dx} = log(x + \frac{1}{x} ) + x \frac{1}{x + \frac{1}{x} } \times 1 - \frac{1}{x {}^{2} } \\ \\ \frac{dy}{dx} = y( log(x + \frac{1}{x} ) + \frac{x {}^{2} - 1 }{x {}^{2} + 1} ) \\ \\ \frac{dy}{dx} = (x + \frac{1}{x} ) {}^{x} ( log(x + \frac{1}{x} ) + \frac{x {}^{2} - 1}{x {}^{2} + 1} )$