The Derivative in Function Analysis

We've established so far that the Derivative is the rate of change of a function. We can use this information to our advantage in function analysis.

If the derivative is positive across an interval or at a point, the function is changing by a positive value, as we go across larger values, the function will increase too, as dictated by whatever function this derivative is.

The same goes if the derivative is negative. The rate of change is negative. This means that the function is somehow decreasing where its derivative is negative.

When the derivative is zero, obviously, there is no change. That's why the derivative of the constant is zero. A constant never changes

We can use this information to our advantage, when analyzing functions.

You see, the derivative is never negative. The function never decreases. It does, however, have a point where the derivative is zero. So, for that specific point we can't be sure that it'll always be increasing.
In this case, we just used the derivative to show that the function is monotone. Specifically, monotonically increasing.