# 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 $f'(x)$? Please explain your answer.

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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, $$f'(x) = \phi ' (x) \cdot g( \phi (x) ).$$

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It should be $f(\phi(x))\cdot\phi'(x)$.