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Given $$f(x) = \int_{0}^{\phi (x)} g(t) dt$$ How could we find $f'(x)$? Please explain your answer.

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up vote 11 down vote accepted

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 $f(\phi(x))\cdot\phi'(x)$.

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Welcome Huang. To make your writing into neat looking pictures, you need to learn a bit of LaTeX code. There are LaTeX guides available online but it's also easy to pick it up just by seeing other peoples examples and imitating their code. To see the code that was used to produce an equation, right click over it and select "Show Source". When you type it on your own, place the code in between \$ signs. Also, I'm pretty sure my answer is correct. Please recheck your working. – Ragib Zaman Nov 18 '11 at 10:10
Thanks for your kind advice! – Huang Nov 18 '11 at 10:17

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