May someone please show me how to proof the following:

Let f be differentiable on I. Show that if f is monotone increasing on I, then

f '(x) >= 0 for all x in I.

Note: >= is greater or equal to.

My idea:

Suppose that f is monotone increasing, that is f(x) <= f(c) for all x such that x<c. ... Then I know that after, we have to somehow say that [f(x)-f(c)]/(x-c) >= 0.

Can someone explain more and do we need any theorem to support the proof?

Thank you .