How to approach integrals as a function? I'm trying to solve the following question involving integrals, and can't quite get what am I supposed to do:
$$f(x) = \int_{2x}^{x^2}\root 3\of{\cos z}~dz$$
$$f'(x) =\ ?$$
How should I approach such integral functions? Am I just over-complicating a simple thing?

Using the Leibnitz rule of differentiation of integrals, 
which states that if
\begin{align}
f(x) = \int_{a(x)}^{b(x)} g(y) \ dy,
\end{align}
then
\begin{align}
f^{\prime}(x) = g(b(x)) b^{\prime}(x) - g(a(x)) a^{\prime}(x).
\end{align}
Thus, for your problem $a^{\prime}(x) = 2$ and $b^{\prime}(x) = 2x$ and, therefore, 
\begin{align}
f^{\prime}(x) = \int_{2x}^{x^2} \sqrt[3]{\cos z} dz = \sqrt[3]{\cos (x^2)} (2 x) - \sqrt[3]{\cos (2x)} (2).
\end{align}
