How do I differentiate this product containing an exponential? I have this equation:
$g(x) = (x+2\sqrt{x})e^x$
If I differentiate this using product and exponent rules for \exp(x)$, is this right:
$g'(x) = (x+2\sqrt{x})e^x + e^x(1+x^{-1/2})$
First * derivative of the second + second * derivative of the first?

That correct, you have correctly applied
$$g(x)=f(x)h(x)\implies g'(x)=f'(x)h(x)+f(x)h'(x)$$
In this case $f(x)=e^x=f'(x)$ so
$$g'(x)=f'(x)h(x)+f(x)h'(x)=[h(x)+h'(x)]f(x)$$
