What is the point of a derivative?

A derivative is a rate of change. For example, let y equal some function of x. Then the derivative of y with respect to x is asking how much y changes with respect to x. We can show this for any x and if we substitute a value in for x then we are showing the rate of change of y when x is equal to that value.
Another way to look at it is, if we had a curve, the derivative of y with respect to a certain x is gradient of the tangent that touches y at the point x. The gradient is showing the rate of change of y at the point x.