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.

the definition of the average rate of change
an example find the average rate of change
In general suppose an object moves along a straight line according to an equation of motion
If a billiard is dropped from a height of Find the average velocity over the intervals

Instantaneous Rate of Change

The difference quotient is the average rate of change of We interpret the limit of the average rate of change as the interval becomes smaller and smaller to be the instantaneous rate of change
given a function Find the instantaneous rate of change at
find the average rate of change and then find the instantaneous rate for change

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.

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.

Find an equation of the tangent line that passes through the points
find an equation of line that oasses through anohter set of points
repeat the process of finding a secant line that gives the tangent line equation

Using differentiation to find the equation of tangent line

the equation of the tangent line at a point and the diagram to understand this theorem
example find the equations of the tangent lines to the given curve that are parallel to a given line
how many tangent lines to a given curve that passes through a given point and where are they tangent
find equations of both tangent lines through the given point that are tangent to the given curve

Horizontal Tangent Lines

if the derivative is zero at a given number then the equation of the tangent line to the curve is a horizontal tangent line
for what values does the graph of a fucntion have a horizontal tangent
To find the horizontal tangent lines we find where the derivative is zero

Relative Rate of Change

Next we illustrate the importance of the relative rate of change as compared to the difference between the absolute rate of change and the average rate of change
Temperature readings in degrees Celsius were recorded every hour starting at midnight on a day in April
Find the average rates of change of temperatures with respect to time and Estimate the instantaneous rate of change at noon

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 definition of the relative rate of change and then find the relative rate of change
Often we are more interested in the relative rate of change of a quantity instead of the instantaneous rate of change
Find the average rate of change find the instantaneous rate of change and find the relative rate of change

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.

A limiting process can be used in the study of curves in general but the derivative the primary 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.

The process of finding the derivative is called differentiation

Taking the Derivative Using the Definition

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

example finding the derivative using the definition of the derivative of a cubic function at a point
example finding the derivative using the definition of the derivative of a cube root function at a point

Finding an Equation of the Tangent Line

the equation of the tangent line theorem and its proof
example Differentiate the function
Find an equation of the tangent line to the graph of

Examples of Non-differentiable Functions

This type of example where the function is not differentiable is called a corner point
This type of example where the function is not differentiable is called a vertical tangent
In this third example does this function have a corner point or is it a vertical tangent

Differentiability Implies Continuity

If a function is differentiable at a number then it is continuous at this number

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.

the basic rules of differentiation sum rule difference rule product rule quotient rule
use differentiation to find the derivative of the function
find the derivative of the function using differentiation
find the derivative of the function using differentiation rules

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.

find the derivative of the function using the differentiation rule called the product rule
find the derivative of the function using the product rule

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.

find the derivative of the function using the quotient rule
find the derivative of the function using the quotient rule and also try again using the product rule

Higher Order Derivatives

second derivative third derivative fourth derivative notation higher order derivatives
Find the first second and third derivatives of a given function

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.

Differentiation Formulas for Trig Functions

differentiation rules for the six trigonometric functions

Derivatives of Trigonometric Functions and Simplification

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

use differentiation rules to find the derivative of a function involving the sine function and the cosine function
find the derivative of a function involving the tangent function

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.

The position of a free falling body neglect air resistance under the influence of gravity can be represented by the function
A ball is thrown vertically upward from the ground with an initial velocity When will it hit the ground With what velocity will the ball hit the ground reach its maximum height what is maximum height
definition of position speed and velocity in terms of the derivative
A particle moving along the horizontal axis has a position given by a function how do you find the total distance traveled

Exercises on Differentiation

Exercises on Differentiation
Exercises on Differentiation
Exercises on Differentiation
Exercises on Differentiation
Exercises on Differentiation
Exercises on Differentiation
Exercises on Differentiation
Exercises on Differentiation
Exercises on Differentiation
Exercises on Differentiation
Differentiation Exercises
Differentiation Exercises
Differentiation Exercises
Differentiation Exercises
Differentiation Exercises
Differentiation Exercises
Differentiation Exercises
Differentiation Exercises
Differentiation Exercises
Differentiation Exercises
Differentiation Exercises

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.

Have fun in your studies!