# Section 11.3

## 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 (x) with respect to x at  is also called the derivative of (x) at x = a.

The derivative of (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.

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