# Differentiation (The Derivative as a Function)

by Dave
(DAVE)—

Motivating the concept of the derivative is an essential step in a student’s calculus education. This article goes through this definition carefully and with several examples allowing a beginning student to absorb the information. Finding the tangent line equation is then detailed, followed by an important theorem: differentiability implies continuity. There’s so much more in this article.

In this article (and accompanying video), I discuss the process of finding the derivative of a function, that is, differentiation. I start with the question, “what is a tangent line, and why does it matter?” and I lead you to the question, “how does the derivative make a difference in the world around us?” Let’s get started.

## From Tangent Lines to Differentiation

So what exactly is a tangent line? Is it a line that only once crosses the graph of a function? No! The tangent line concept is more subtle than that. In this article, I discuss the average rate of change, the instantaneous rate of change, what exactly is a tangent line. I also cover the relative rate of change.

The importance of the tangent line is motivated through examples by discussing average rate of change and instantaneous rate of change. We place emphasis on finding an equation of a tangent line especially horizontal line tangent lines. At the end we consider relative rates of change.

### Average Rate of Change

We begin with the average rate of change of a function over a closed interval.

### What is a Tangent Line?

The tangent line problem is widely considered to be the instigating idea behind the derivative. Computing the slope of a tangent line was a problem that the French mathematician Pierre de Fermat developed. Picking up on these ideas were Isaac Newton and Gottfried Liebniz, who then developed differential calculus.

For a general curve it is not easy to define what is meant by a tangent line; for example a tangent line might mean a line touches the curve only once, but this does not work in all cases.

The important idea to remember is that a tangent line is a local concept, we say the tangent line at a point.

In our first example we will calculate a series of functions whose graphs are secant lines to the graph of a given function and use them to infer an equation of the tangent line at a point.

### Relative Rate of Change

Sometimes we are not interested in the instantaneous rate of change and instead we may want a relative rate of change (percentage). For example suppose a student makes a 39 on a test, this would be a very good grade if the score is out of 40 points. However if the score was out of a total of 100 points then the grade is not so good.

## The Derivative as a Function

We begin with the definition of the derivative as a limit of a difference quotient. We then give several examples of how to find the derivative of a function using this definition. Finding an equation of the tangent line is then considered, and after several examples of this, we then give examples of how a function may not be differentiable. At the end we prove that every function that is differentiable at a point must also be continuous at that point.

### The Definition of Derivative

The first main idea of calculus is of course, the limit. A limiting process can be used in the study of curves in general; but the derivative is the main limiting process that has lead to the development of calculus.

The limiting process illustrated in the examples below was first developed by the French mathematician Pierre de Fermat. The following definition was realized by Newton and Leibniz.

### Taking the Derivative Using the Definition

In the following examples we illustrate how to find the derivative function using the definition of the derivative.

## Differentiation Rules (with Examples)

This section discusses the linearity rule for derivatives and its special cases, such as the sum and difference rules. Then I explain what higher-order derivatives are, including the notation in both forms.

Computing the limit of the difference quotient can be tedious and require ingenuity; fortunately for a large number of common functions there is a better way to compute the derivative. In this section, we detail the power rule and the linearity rule for differentiation. These rules greatly simplify the task of differentiation. We also give examples on how to find the tangent line given some geometric information and to find the horizontal tangent line(s) to the graph of a given function.

### Differentiation Rules

We begin with a theorem states the common procedural rules for taking derivatives. For example, the derivative of a sum of functions is the sum of the derivative functions. The same is not true for a product of functions. To convince yourself that the derivative of the product of two functions is not the product of the derivative functions try to find a counterexample.

### Product Rule and Quotient Rule

Maybe you are wondering what the product rule is, or are you are trying to get a handle on using the quotient rule? I discuss several examples that will help you understand these theorems. I work through them slowly and show you how to simplify. After you master these rules, you will be ready for more advanced rules, such as the chain rule.

#### The Product Rule

We being with the product rule for find the derivative of a product of functions. The rule can be generalized to more than the product of two functions.

### The Quotient Rule

The quotient rule is a theorem for finding the derivative of a function which can be written as the ratio of two differentiable functions.

## Derivatives of Trigonometric Functions

Perhaps, well, do you know that the derivative of sine is cosine? Next, I go through the derivatives of the six trigonometric functions. One-by-one, we see how to use the product rule and the quotient rule to figure out these derivatives. But I start it all with the derivative of the sine function using the definition of the derivative.

Formulas for finding the derivative of the six trigonometric functions are given. We assume that the trigonometric functions are functions of real numbers (angles measured in radians) because the trigonometric differentiation formulas rely on limit formulas that become more complicated if the degree measurement is used instead of radian measure.

### Derivatives of Trigonometric Functions and Simplification

Since the trigonometric functions are differentiable functions on their domains they are also continuous functions on their domain.

## Applications of Differentiation (The Role of the Derivative)

Derivatives have many applications throughout the sciences. But every student needs their first examples of this critical concept. Next, I go through some beginning examples illustrating the importance of the derivative. Specifically, I examine the free-falling body problem.

### Free-Falling Body

Basically, rectilinear motion refers to the motion of an object that can be modeled along a straight line; and the so-called falling body problems are a special type of rectilinear motion where the motion of an object is falling (or propelled) in a vertical direction. Another type of rectilinear motion is the free-falling body problem.

## Calculus 1 (Explore, Discover, Learn) Series

If you would like me to make a video with the solutions to some of the exercises let me know in the comments.

This article (and accompanying video) are a part of a series of articles (and videos) called the Calculus 1 (Explore, Discover, Learn) Series. Also, I put together for you a getting started with calculus 1 page and a video playlist for calculus one.