# The Definition of a Derivative

The graphical idea of a **slope of a tangent line** is very useful, but for some uses we need a more algebraic definition of the **derivative of a function**. We will use this definition to calculate the derivatives of several
functions and see that the results from the definition agree with our graphical understanding. We will also look at several different interpretations for the derivative, and derive a theorem which will allow us to easily and quickly determine the
derivative of any fixed power of .

In the last section we found the slope of the tangent line to the graph of the function at an arbitrary point by calculating the slope of the secant line through the points and ,

and then by taking the limit of as h approached 0 (Fig. 1). That approach to calculating slopes of tangent lines is the definition of the derivative of a function.

**Definition of the Derivative:**

The **derivative **of a function is a new function, ' (pronounced "eff prime"),

whose value at is if the limit exists and is finite.

This is the definition of differential calculus, and you must know it and understand what it says. The rest of this chapter and all of Chapter 3 are built on this definition as is much of what appears in later chapters. It is remarkable that such a simple idea (the slope of a tangent line) and such a simple definition (for the derivative ) will lead to so many important ideas and applications.

**Notation:** There are three commonly used notations for the **derivative of **:

emphasizes that the derivative is a **function **related to

emphasizes that we perform an **operation **on to get the derivative of

We will use all three notations so you can get used to working with each of them.

f' represents the slope of the tangent line to the graph of at the point or the instantaneous rate of change of the function at the point ).

If, in Fig. 2, we let be the point , then . As , we see that and so

We will use whichever of these two forms is more convenient algebraically.

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.2-Definition-of-Derivative.pdf

This work is licensed under a Creative Commons Attribution 3.0 License.