Question

Differentiate the following functions with respect to $x$
${(1+\frac{1}{x})}^x$

Answer 1

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

Taking log on both sides

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

Differentiate both sides w.r.t x

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

$\frac{dy}{dx}=y[\log \left(\frac{1+x}{x}\right)-\frac{1}{1+x}]$

$\frac{dy}{dx}={(1+\frac{1}{x})}^x[\log \left(\frac{1+x}{x}\right)-\frac{1}{1+x}]$