# Proof Using Mean Value Theorem

Hello all

Using the Mean Value Theorem, prove that the derivative of the indefinite integral $\int f(x) \ dx$ is $f(x)$

So do I just use the fact that $\int^b_a f(x) \ dx = f(\xi)(b-a)$?

Thanks

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is this right?

HallsofIvy
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You CAN'T use the mean value theorem to prove the Fundamental Theorem of Calculus.

not prove but maybe show

hey... do you forget our rule here... don't give out the answer... delete the link and give him some hints lead to the answer

learningphysics
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vincentchan said:
hey... do you forget our rule here... don't give out the answer... delete the link and give him some hints lead to the answer

learningphysics
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I believe your question should to prove that the derivative of this function:

$$F(x)=\int_a^{x} f(t)dt$$

is f(x). Am I right? The above is a definite integral. The derivative of the indefinite integral is just f(x) by definition. Indefinite integral means anti-derivative.

What is F'(x) from first principles i.e: using the definition of derivative?

$$F'(x) = \frac{F(x+\Delta x) - F(x)}{\Delta x}$$

forgot to put limit

Last edited:
learningphysics
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$$F'(x) = \frac{f(x+\Delta x) - f(x)}{\Delta x}$$

forgot to put limit
Careful... we're looking for F'(x) not f'(x).

is this right?

learningphysics
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