IF $f(x) = \int_{0}^{\phi (x)} g(t) dt$, How could we find $f'(x)$? Given $$f(x) = \int_{0}^{\phi (x)} g(t) dt$$ How could we find $f&#39;(x)$? Please explain your answer.
Assume that $\phi (x) $ is a differentiable function. The right hand side is the composition of two simpler functions, each of which we can find the derivative. 

$$ f(x) = G( \phi (x) ) $$ where $ G(x) =\displaystyle \int^x_0 g(t) dt.$ By the chain rule, $$ f&#39;(x) = \phi &#39; (x) \cdot G&#39;( \phi (x) ) .$$

By the Fundamental Theorem of Calculus, we have $ G&#39;(x) = g(x) $ so $ G&#39;( \phi (x) ) = g( \phi (x) ).$

Thus, $$ f&#39;(x) = \phi &#39; (x) \cdot g( \phi (x) ). $$
It should be $f(\phi(x))\cdot\phi&#39;(x)$.