I don't understand what you are doing! You say that the problem is to find the derivative but you seem to be trying to do an itegral.
First, you understand that it is not easy to turn a computer monitor on it side! But I think your function is \(\displaystyle 4x+ \int_{x^2}^{x^3}e^{t^2}dt\). That integral cannot be done in terms of elementary functions, and this problem does not require you to itegrate it.
You can, as I think you are trying to do, write this the integral as \(\displaystyle \int_0^{x^3}e^{t^2}dt- \int_0^{x^2}e^{t^2}dt\) and do each separately:
Write \(\displaystyle u= x^3\) so that the first itegral becomes \(\displaystyle \int_0^{u} e^{t^2}dt\) and its integral, with respect to u, by the "Fundamental Theorem of Calculus", is \(\displaystyle e^{u^2}\) so that the derivative with respect to x is \(\displaystyle e^{u^2}\frac{du}{dx}= e^{(x^3)^2}(3x^2)\)
More general is "Leibniz's rule":
\(\displaystyle \frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} f(x,t)dx= f(x,\beta(x)\frac{d\beta}{dx}- f(x, \alpha(x))\frac{d\alpha}{dx}+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial f}{\partial x}dt\)
