## The Derivative at a Point

In section 11.2, we introduced the idea of the instantaneous rate of change of a function. This idea is critical to understanding how a quantity is changing with respect to another. The instantaneous rate of change of a function *f *(*x*) with respect to *x* at is also called the derivative of *f *(*x*) at *x* = *a*.

The derivative of *f *(*x*) at *x* = *a* is defined as

provided the limit exists. The symbol *f* ′(*a*), is read “*f *prime of *a*”.

In this section we’ll look at the derivative of a function from a geometric viewpoint by examining slopes of secant lines and how they can be used to find the slope of a tangent line. We will also find the derivative of a function at a point. This is essentially the same process we used to calculate the instantaneous rate of change of a function given by a formula at a point.

Once we have looked at the idea of a derivative geometrically and have taken the derivative of several functions, we will explore what a derivative of a function tells us for several business functions.

**Read **in Section 11.3

- What is a derivative?
- Handout: Secant Line Versus Tangent Line
- How do you compute the derivative at a point using a limit?
- How can you use a tangent line to forecast function values?
- What does the derivative at a point tell you about a function?

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