Proof that f(x)=F'(x)...that derivative of antiderivative equals original function
2,143
This is true by definition. We say a function $F(x)$ is an antiderivative of $f(x)$ when $F'(x) = f(x)$.
Related videos on Youtube
Author by
Muhammad Umer
Updated on March 17, 2020Comments

Muhammad Umer over 3 years
Is there any proof for this as far i can find fundamental theorem is used to proof this...And fundamental theorem is proven using this.
So to me it sounds like chicken egg thing...
I have been doing this whole day...
http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#cite_note5
in link above it says that since
f(x)= A'(x) therefore A(x)=F(x);
and when i go to understand why F'(x) = f(x)...or in this Case How antiderivative of integral A equals A. I get referenced back to fundamental theorem. Is it using itself as a proof?

TooTone over 9 yearsdoes this help? math.stackexchange.com/questions/663487/…

TooTone over 9 yearssee also en.wikipedia.org/wiki/…

Muhammad Umer over 9 yearswikipedia does the same thing. It assumes integral = area and then shows F'(x)=f(x)....If you want to know why area = integral? you get answer like well f(x) = A'(x) so F(x)=A(x).?????

Muhammad Umer over 9 yearsforexample: at line where it says
According to the mean value theorem for integration, there exists a real number
it makes integral equal to area derived from mean value theorem. But Why. For to proof that integral equals area it needs to be proven that derivative of integral equals original function. Because what can be proven is that original function f(x) = A'(x). 
augurar over 9 years@MuhammadUmer The mean value theorem for integration does not require the fundamental theorem of calculus.


Muhammad Umer over 9 yearsif it's true by definition then why did anyone bother with 1st fundamental theorem of calculus.

Muhammad Umer over 9 yearsWithout fundamental theorem F(x) is just notation for Riemann sum and derivative is something else separate.

augurar over 9 years@MuhammadUmer You are getting confused by the notation. Sometimes, $F(x)$ is used to mean an antiderivative of $f(x)$. Other times, $F(x)$ is defined as $\int_a^x f(t) dt$. You will have to read the text to determine which one of these is meant in a particular case.

Muhammad Umer over 9 yearsHere i meant it as first one.

augurar over 9 years@MuhammadUmer Then the claim is true by definition.

Muhammad Umer over 9 yearsIt's as you say..it took me longest time but i have got it. F(x) if meant as integral it only means the riemann sum before Fundamental Theorem proofs it as more...Basically...riemann sum does equal area under curve and integral is its another form. But FTC using mean value theorem shows that for F'(x) there exists a function, call it g(x). It means that riemann sum has a derivative (a function...), which means There is a function of which antiderivative is This riemann sum. (it's as you say derivative and antiderivative are related by definition).

Muhammad Umer over 9 yearsSO when we see F'(x) = f(x)..it is not really saying that derivative of antiderivative equals main function but that derivative of riemann sum equals some function. But since antiderivative exists as well then that means riemann sum itself also equals that antiderivative function..as derivative of that antiderivative would be this function. therefore in end it does end up meaning what i said it doesn't mean.
F'(x) = g(x) And also g(x) = G'(x) G is antiDeriv...that means G = F
so antiderivative of g is the area.G === AREA