How do I differentiate this product containing an exponential?

Question

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?

Answer 1

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

$$g'(x)=f'(x)h(x)+f(x)h'(x)=[h(x)+h'(x)]f(x)$$