How to find derivative of an integral of this type $$f(x) = \int _x^{e^x}\:\left(\sin t^2\right)\,dt$$
How to find the derivative $f'(x)$
Attempt: $\sin (e^{x^2})  e^x$

Use Fundamental theorem of calculus, let $F$ be antiderivative of $\sin t^2$, then you have:
$$f(x)=F(e^x)-F(x)$$
So:
$$f'(x)=e^xF'(e^x)-F'(x)=e^x \cdot \sin ((e^{x})^2)-\sin (x^2)$$
