TITLE: IF $f(x) = \int_{0}^{\phi (x)} g(t) dt$, How could we find $f'(x)$? 
QUESTION [5 upvotes]: Given $$f(x) = \int_{0}^{\phi (x)} g(t) dt$$ How could we find $f'(x)$? Please explain your answer.

REPLY [11 votes]: 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, $$ f'(x) = \phi ' (x) \cdot G'( \phi (x) ) .$$
By the Fundamental Theorem of Calculus, we have $ G'(x) = g(x) $ so $ G'( \phi (x) ) = g( \phi (x) ).$
Thus, $$ f'(x) = \phi ' (x) \cdot g( \phi (x) ). $$