How does the first fundamental theorem of calculus guarantee the existence of antiderivatives of functions?

First fundamental theorem of calculus: $$g(x) = \int_a^xf(t)dt$$ then $$g'(x) = f(x)$$

But how does this guarantee the existence of antiderivatives of functions? Tutorials always state it does, but never explain why.

Also, once we know it guarantees the existence of antiderivatives of functions, does this mean we can always use the second fundamental theorem (where you split a definite integral into two indefinite ones)? Is that the key purpose to realise here?

If $f$ is an arbitrary continuous funciton, then you can define a function $g$ via the integral and by the fundamental theorem, the derivative of $g$ is $f$. That is: $g$ is an antiderivative of $f$. That's all.

• If I were to visualize it, how does taking the derivative of an area under a curve give me the original function? I just don't get this... Is there some logic behind it, or am I supposed to just leave it be and think about it as being true by definition? – user1534664 Apr 20 '15 at 21:12
• @user1534664 maybe it will help you to look at the proof of the theorem, and to create graphs of each step e.g. when we invoke the MVT. – GFauxPas Apr 20 '15 at 21:51
• @GFauxPas I actually get it now. I don't know how I was that stupid, just tired I guess :) – user1534664 Apr 21 '15 at 21:19
• Can anyone answer my second question? I don't think opening another question just for that is benefitial to anyone but me. – user1534664 Apr 21 '15 at 21:24
• @user1534664 You're not stupid; I had the same difficulty you had when I first learned Calculus – GFauxPas Apr 21 '15 at 22:56

If you read it in the other direction, then for a function $g$, the anti-derivative is $$G(x) = \int_a^x g(t) dt + c.$$

• an anti derivative – Henry Apr 20 '15 at 21:36
• You're right, of course. – Andre Apr 20 '15 at 21:41
• Can anyone answer my second question? I don't think opening another question just for that is benefitial to anyone but me. – user1534664 Apr 21 '15 at 21:24
• @user1534664 Of course it will benefit other people! Don't think you're the first or last person to have trouble with the FToC! – GFauxPas Apr 21 '15 at 22:59