The derivative

The derivative measures how fast a function changes. On a graph, it is the slope of the tangent line. In an application, it is the instantaneous rate of change, such as the velocity of a moving object.

In this chapter, you will learn where the derivative comes from, how to find it from its limit definition, and what it tells us about a graph.

Finding derivatives with limits takes many steps. The next chapter gives rules that make it much faster.

This chapter covers the following topics:

Tangent lines and rates of change

Secant lines and tangent lines, the slope of a tangent line as a limit, average and instantaneous velocity, and average and instantaneous rates of change.

The derivative at a point

The definition of the derivative, what it means, finding it with the difference quotient, reading it from a graph, its units, and a function that is not differentiable.

The derivative as a function

The derivative as a new function, derivative notation, the graphs of a function and its derivative, where a function is not differentiable, and why differentiable functions are continuous.

Before you start

This chapter builds on limits and continuity. These pages may help:

• Computing limits (factoring, the conjugate, combining fractions, expanding)
• One-sided limits
• Continuity
• Slope of a line
• Equations of a line (point-slope form)