First and second derivatives of the function $f(x)=x\int_0^x e^{t^2}dt$

Question

I haven't done calculus for a while so I need your help with these two exercises. I am not sure whether my solutions are correct so I'd really appreciate someone's feedback.

$$ f(x)=x\int_0^x e^{t^2}dt $$ Find $\frac{d}{dx}f(x)$ and $\frac{d^2}{dx^2}f(x)$

Here is a link to exercises.

Answer 1

you can use the formula $$\frac{d}{dx}\int_{f(x)}^{g(x)} h(t) dt=g'(x)h(g(x))-f'(x)h(f(x))$$ so we have that $f'=\int_{0}^{x} e^{t^2} dt+xe^{x^2} $ and $f''=2e^{x^2} +2x^2e^{x^2}$