Question

differentiate ex(logx+1/x)

Answer 1

Use the product rule of differentiation,

$\frac{d(f(x)g(x))}{dx} =f'(x)g(x)+f(x)g'(x)$.

Let $f(x)=e^x,g(x)=\ln x+\frac{1}{x}$

The derivative of $e^x(\ln x+\frac{1}{x} )$ is

$\frac{d}{dx} [e^x(\ln x+\frac{1}{x} )]=e^x(\ln x+\frac{1}{x} )+e^x(\frac{1}{x} -\frac{1}{x^2} )\\ \frac{d}{dx} [e^x(\ln x+\frac{1}{x} )]=e^x(1+\frac{2}{x} -\frac{1}{x^2} )$