Hey guys this was given to me as an exercise question and its really confusing. I'm not really sure where to start with this one, and I am assuming that the derivative isn't just $e^{-t^2} dt$. Anyways, any help is appreciated, thank you!.
Find the derivative of $$\int \limits_x^{x^2} e^{-t^2}dt $$
Best Answer
I assume you mean find the derivative of $F(x)$, where
$$F(x) = \int_x^{x^2} e^{-t^2} dt.$$
Let $G(x) = \int_0^x e^{-t^2} dt $. By the fundamental theorem of calculus,
\begin{align*} F'(x) &= \frac{d}{dx}(G(x^2) - G(x)) \\ &= G'(x^2)(2x) - G'(x) \\ &= e^{-x^4}(2x) - e^{-x^2}. \end{align*}