I have trouble doing proofs using the fundamental theorem of calculus and I think seeing an example would help.
Suppose we have a continuous function $ f $ defined on a real interval and a function $ F $ that is also defined on the same interval; and $F(x)=\int^b_x fdt$. Then for $ x $ in that interval, $ F'(x)=-f(x) $.
I know this can be done using the fundamental theorem of calculus; how is it applied?
By the way, the theorem says this:
Let $ f $ be integrable on $[a,b]$. For $x\in[a,b]$ put $F(x)=\int^x_a f(t)dt$. Then $ F$ is continuous on $[a,b]$ and furthermore if $ f $ is continuous at a point $x_0$ of $[a,b]$ then $F$ is differentiable at $x_0$ and $F'(x_0)=f(x_0)$.