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.