Answer
$$ f'(x)= 2x(1+x)e^{2x}.$$
Work Step by Step
Recall the product rule: $(uv)'=u'v+uv'$
Recall that $(e^x)'=e^x$
Since we have
$$ f(x)=x^2 e^{2x}$$
then the derivative $ f'(x)$, using the chain and product rules, is given by
$$ f'(x)= (x^2)'e^{2x}+x^2(e^{2x})'=2xe^{2x}+2x^2e^{2x}=2x(1+x)e^{2x}.$$
