Given $f(x) = \int_{0}^{\phi (x)} g(t) dt$ How could we find f'(x)? Please explain your answer.
IF $f(x) = \int_{0}^{\phi (x)} g(t) dt$, How could we find $f'(x)$?
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calculus
2 Answers
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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).
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1Thanks for your kind advice! – 2011-11-18