# The Derivative

### Introduction and Summary

The derivative is the first of the two main tools of calculus (the second being the integral). The derivative is the instantaneous rate of change of a function at a point in its domain. This is the same thing as the slope of the tangent line to the graph of the function at that point. In order to give a rigorous definition for the derivative, we need the concept of limit introduced in the preceding section.

Given a function f , we can define a derivative function f' to take on the value of the derivative of f at each point in the domain. For example, if Otis drives in a straight line from his home to Grand Rapids, Michigan, and the function f (t) gives his distance from home at time t , then the function f'(t) gives his "instantaneous rate of change", or his velocity, at time t .

Once we have taken the derivative of a function f once, we can take the derivative again. This is called the second derivative of the original function f , and equals the "instantaneous rate of change of the instantaneous rate of change" of f . In the example above, this corresponds to how quickly Otis is speeding up or slowing down, that is, his acceleration. We can continue in this manner as long as we like, taking successive derivatives.

In this SparkNote, we define derivatives and seek to develop an intuitive understanding of their meaning. In the following chapters, we will see how to compute derivatives and will explore some of their many applications.