Question

Differentiate ${(x+\frac{1}{x})}^x$ $w.r.t$ $x$.

Answer 1

Let $y={(x+\frac{1}{x})}^x$

$\Rightarrow \log y=\log {(x+\frac{1}{x})}^x$

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

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

$\Rightarrow \frac{1}{y}\frac{dy}{dx}=\log (x+\frac{1}{x})+x\frac{1}{x+\frac{1}{x}}(1-\frac{1}{x^2})$

$\Rightarrow \frac{dy}{dx}=y\log (x+\frac{1}{x})+y\frac{x^2}{x^2+1}\frac{x^2-1}{x^2}$

$\Rightarrow \frac{dy}{dx}=y\log (x+\frac{1}{x})+y\frac{x^2-1}{x^2+1}$

$\Rightarrow \frac{dy}{dx}={(x+\frac{1}{x})}^x\log (x+\frac{1}{x})+{(x+\frac{1}{x})}^x\frac{x^2-1}{x^2+1}$