How to prove if a function is increasing, using derivatives?

First, we'll choose 2 numbers: x1<x2.

Based on the rule of an increasing function, we'll have to prove that f(x1)<f(x2), also.

Supposing that is given the fact that f'(x)>0 and from here, we'll conclude that f is differentiable on the interval [x1, x2].

Now, we'll apply the Mean Value Theorem, which states that:

f(x2)-f(x1)=f'(c)(x2-x1)

From enunciation, we have that f'(c)>0 and, because we've choosen that x2>x1, we'll havex2-x1>0.

So, the product f'(c)(x2-x1)>0.

But the product f'(c)(x2-x1) = f(x2)-f(x1)

From here, we conclude that f(x2)-f(x1)>0

That means that f(x2)>f(x1), which means that f(x) is an increasing function.