# The Derivative of a Function

The concept of the derivative of a function is central to Calculus - it is the reason for the subject. In an intuitive sense, the derivative of a function at any point is the rate of change of that function. For example, if $$p$$ is a function that describes your distance from your house at any time, then the derivative of $$p$$ describes your speed at that time. Or again, if $$h$$ is a function that describes your height on a ski lift in terms of distance from the terminal, then the derivative of $$h$$ would roughly give the steepness of the slope at that distance up the lift.

It turns out that the derivative of a function provides an easy way for us to find a polynomial approximation to that function. It is very unconventional to approach the derivative this way, but that's what we are going to do.