Question

Let $y=x{e}^x$ then prove that $x\frac{dy}{dx}=(1+x)e^x$.

Answer 1

Given,

$y=x{e}^x$

Now differentiating both sides with respect to $x$ we get,

$\frac{dy}{dx}=x{e}^x+e^x$

or, $x\frac{dy}{dx}=x^2{e}^x+x{e}^x$

or, $x\frac{dy}{dx}=(x+1)e^x$