Explainer: Definition of the Derivative

In this explainer, we will learn how to calculate the derivative of a function using the formal definition of the derivative as a limit.

Rates of change form a central concept in many real-world problems from mechanics to reaction rate and from modeling population growth to modeling radioactive growth and decay. In fact, the development of Newtonian mechanics is intimately connected to the mathematical study of rates of change. In many ways, one of the large contributing factors to the development of the study of rates of change was to be able to mathematically describe the mechanics of the real world.

If we consider a car accelerating from a position of rest, we see that the speed increases. Initially, we have a speed of zero; then, we might be going 10 miles per hour and then 20. However, what does it mean to say that we are traveling at a particular speed? For example, when you see that that speedometer reads 10 miles per hour, what does this mean? We might say that if we did not change our speed, we would travel 10 miles in an hour. Or we might consider how many meters we travel in a second, or maybe a smaller interval of time. However, clearly, if we consider how far we travel in zero time, it would be zero meters. In addition to all of this, we are not traveling with constant speed; we are accelerating.

The issue of trying to make these things mathematically rigorous is the challenge that the great mathematicians of the past such as Newton and Leibniz formalized for us as derivatives. In this explainer, we will introduce the definition of a derivative and how this enables us to deal with the apparent contradiction of talking about an instantaneous rate of change.

Conceptually, we can think of the idea of a derivative at a point π‘₯ as taking better and better approximations to the tangent of a curve at a given point. We can do this by considering the slopes of secant lines going π‘₯ and a second point on the curve as this point approaches π‘₯.

Clearly, as β„Ž gets closer to 0, the approximation to the slope of the tangent gets better and better. The derivative is formally defined as the limit of this process.

Definition of the Derivative

The derivative of a function at a point π‘₯ is defined as limο‚β†’οŠ¦οŠ¦οŠ¦π‘“(π‘₯+β„Ž)βˆ’π‘“(π‘₯)β„Ž, where this limit exists. An alternative but equivalent definition of the derivative at π‘₯ is limο—β†’ο—οŠ§οŠ¦οŠ§οŠ¦οŽ οŽŸπ‘“(π‘₯)βˆ’π‘“(π‘₯)π‘₯βˆ’π‘₯, if the limit exists.

There are two common ways we denote derivatives: Leibniz’s notation and prime notation (sometimes referred to as Lagrange’s notation). For a function 𝑦=𝑓(π‘₯), Leibniz’s notation for the derivative is written using the infinitesimals d𝑦 and dπ‘₯ as dd𝑦π‘₯, which we read as β€œthe derivative of 𝑦 with respect to π‘₯ ” or β€œd𝑦 by dπ‘₯.”

Using prime notation, the derivative of 𝑓(π‘₯) with respect to π‘₯ is denoted 𝑓(π‘₯) which we read as β€œπ‘“ prime of π‘₯.”

It is important for the student of calculus to be comfortable using either of the definitions of the derivative and with both forms of notation.

Often, using the first definition of the derivative is easier; however, in some cases it is just as simple to use either form. In the first example, we will demonstrate using the second form of the definition of the derivative.

Example 1: Calculating the Derivative at a Point

Find the derivative of 𝑓(π‘₯)=π‘₯ at the point π‘₯=2 from first principles.

Answer

Recall the definition of the derivative at π‘₯: 𝑓(π‘₯)=𝑓(π‘₯)βˆ’π‘“(π‘₯)π‘₯βˆ’π‘₯.οŽ˜οŠ¦ο—β†’ο—οŠ§οŠ¦οŠ§οŠ¦lim

Letting π‘₯=2 and 𝑓(π‘₯)=π‘₯, we have 𝑓(2)=(π‘₯)βˆ’(2)π‘₯βˆ’2=π‘₯βˆ’4π‘₯βˆ’2.οŽ˜ο—β†’οŠ¨οŠ§οŠ¨οŠ¨οŠ§ο—β†’οŠ¨οŠ¨οŠ§οŠ§limlim

Notice that the numerator is the difference of two squares that we can factor as follows: 𝑓(2)=(π‘₯βˆ’2)(π‘₯+2)π‘₯βˆ’2.οŽ˜ο—β†’οŠ¨οŠ§οŠ§οŠ§lim

Since π‘₯βˆ’2β‰ 0, we can cancel this common factor in the numerator and denominator to get 𝑓(2)=(π‘₯+2).οŽ˜ο—β†’οŠ¨οŠ§lim

We can now take the limit to get 𝑓(2)=4.

Earlier, we gave the definition of the derivative at a point. However, we can actually consider the derivative as a function whose value is equal to the slope of the tangent to the curve at each value of π‘₯. In the next example, we will demonstrate the process of finding the derivative as a function and using it to find the slope of the tangent.

Example 2: Finding the Slope of the Tangent Using the Derivative

Let 𝑓(π‘₯)=8π‘₯βˆ’6π‘₯+9. Use the definition of the derivative to determine 𝑓(π‘₯). What is the slope of the tangent to its graph at (1,2)?

Answer

We will use the definition of the derivative, 𝑓(π‘₯)=𝑓(π‘₯+β„Ž)βˆ’π‘“(π‘₯)β„Ž,οŽ˜ο‚β†’οŠ¦lim to find an expression for 𝑓(π‘₯). Using the function we have been given, we have 𝑓(π‘₯)=ο€Ί8(π‘₯+β„Ž)βˆ’6(π‘₯+β„Ž)+9βˆ’ο€Ή8π‘₯βˆ’6π‘₯+9ο…β„Ž=8(π‘₯+β„Ž)βˆ’8π‘₯βˆ’6(π‘₯+β„Ž)+6π‘₯β„Ž.οŽ˜ο‚β†’οŠ¦οŠ¨οŠ¨ο‚β†’οŠ¦οŠ¨οŠ¨limlim

Expanding the parentheses in the numerator, we have 𝑓(π‘₯)=8ο€Ήπ‘₯+2π‘₯β„Ž+β„Žο…βˆ’8π‘₯βˆ’6π‘₯βˆ’6β„Ž+6π‘₯β„Ž=16π‘₯β„Ž+8β„Žβˆ’6β„Žβ„Ž.οŽ˜ο‚β†’οŠ¦οŠ¨οŠ¨οŠ¨ο‚β†’οŠ¦οŠ¨limlim

Since β„Žβ‰ 0, we can cancel this common factor from the numerator and denominator to get 𝑓(π‘₯)=(16π‘₯+8β„Žβˆ’6).οŽ˜ο‚β†’οŠ¦lim

We can now take the limit as β„Žβ†’0 which gives us 𝑓(π‘₯)=16π‘₯βˆ’6.

Since the value of the derivative at a point is equal to the slope of the tangent to the curve at that point, the slope of the tangent to the graph at (1,2) is equal to 𝑓(1)=16(1)βˆ’6=10.

Using the definition of the derivative, we can find the derivative of many different types of functions by using a number of algebraic techniques to evaluate the limits. In the next couple of examples, we will use the definition of the derivative to find the derivative of reciprocal and radical functions.

Example 3: Finding the Derivative of a Reciprocal Function

Using the definition of a derivative, evaluate ddπ‘₯ο€Ό11+π‘₯.

Answer

Recall the definition of the derivative: ddlim𝑦π‘₯=𝑓(π‘₯+β„Ž)βˆ’π‘“(π‘₯)β„Ž.ο‚β†’οŠ¦

Setting 𝑦=11+π‘₯, we have ddlim𝑦π‘₯=ο€Όοˆβˆ’ο€»ο‡β„Ž.ο‚β†’οŠ¦οŠ§οŠ§οŠ°(ο—οŠ°ο‚)οŠ§οŠ§οŠ°ο—

We begin by expressing this as a single fraction over a common denominator: ddlimlimlim𝑦π‘₯=ο€Όοˆβˆ’ο€Όοˆβ„Ž=1+π‘₯βˆ’(1+π‘₯+β„Ž)β„Ž(1+(π‘₯+β„Ž))(1+π‘₯)=βˆ’β„Žβ„Ž(1+(π‘₯+β„Ž))(1+π‘₯).ο‚β†’οŠ¦οŠ§οŠ°ο—((ο—οŠ°ο‚))(οŠ§οŠ°ο—)(ο—οŠ°ο‚)((ο—οŠ°ο‚))(οŠ§οŠ°ο—)ο‚β†’οŠ¦ο‚β†’οŠ¦

Since β„Žβ‰ 0, we can cancel it from the numerator and denominator to get ddlim𝑦π‘₯=ο€½βˆ’1(1+(π‘₯+β„Ž))(1+π‘₯).ο‚β†’οŠ¦

We can now use the rules of finite limits to rewrite this as ddlimlim𝑦π‘₯=βˆ’1(1+(π‘₯+β„Ž))(1+π‘₯)=βˆ’1(1+π‘₯)(1+(π‘₯+β„Ž)).ο‚β†’οŠ¦ο‚β†’οŠ¦

Taking the limit as β„Žβ†’0, we have dd𝑦π‘₯=βˆ’1(1+π‘₯).

Example 4: Finding the Derivative of a Radical Function

Find the derivative of the function 𝑔(π‘₯)=βˆ’8βˆšβˆ’π‘₯+9 using the definition of the derivative, and then state the domain of the function and the domain of its derivative.

Answer

Using the definition of the derivative that 𝑔(π‘₯)=𝑔(π‘₯+β„Ž)βˆ’π‘”(π‘₯)β„Ž,οŽ˜ο‚β†’οŠ¦lim we have 𝑔(π‘₯)=ο€Ίβˆ’8βˆšβˆ’(π‘₯+β„Ž)+9ο†βˆ’ο€»βˆ’8βˆšβˆ’π‘₯+9ο‡β„Ž=8βˆšβˆ’π‘₯+9βˆ’8βˆšβˆ’(π‘₯+β„Ž)+9β„Ž.οŽ˜ο‚β†’οŠ¦ο‚β†’οŠ¦limlim

Before we are able to evaluate this limit, we need to multiply the numerator and denominator by the conjugate of the numerator as follows: 𝑔(π‘₯)=8βˆšβˆ’π‘₯+9βˆ’8βˆšβˆ’(π‘₯+β„Ž)+9β„Ž8βˆšβˆ’π‘₯+9+8βˆšβˆ’(π‘₯+β„Ž)+98βˆšβˆ’π‘₯+9+8βˆšβˆ’(π‘₯+β„Ž)+9=64(βˆ’π‘₯+9)βˆ’64(βˆ’(π‘₯+β„Ž)+9)β„Žο€»8βˆšβˆ’π‘₯+9+8βˆšβˆ’(π‘₯+β„Ž)+9=64β„Žβ„Žο€»8βˆšβˆ’π‘₯+9+8βˆšβˆ’(π‘₯+β„Ž)+9.οŽ˜ο‚β†’οŠ¦ο‚β†’οŠ¦ο‚β†’οŠ¦limlimlim

Since β„Žβ‰ 0, we can cancel it from the numerator and denominator to get 𝑔(π‘₯)=648βˆšβˆ’π‘₯+9+8βˆšβˆ’(π‘₯+β„Ž)+9.οŽ˜ο‚β†’οŠ¦lim

We can now use the rules of finite limits to rewrite this as 𝑔(π‘₯)=648βˆšβˆ’π‘₯+9+8βˆšβˆ’(π‘₯+β„Ž)+9.οŽ˜ο‚β†’οŠ¦lim

Taking the limit as β„Žβ†’0, we get 𝑔(π‘₯)=6416βˆšβˆ’π‘₯+9=4βˆšβˆ’π‘₯+9.

We now need to consider the domain of the function and its derivative. Since the square root of a negative number is not a real number, the domain of 𝑔 should be restricted to the values of π‘₯, which results in a nonnegative expression under the square root. Hence, the domain of 𝑔 is defined by βˆ’π‘₯+9β‰₯0.

This is equivalent to π‘₯≀9 which can be written as an interval as (βˆ’βˆž,9]. Clearly, the same restriction applies to the derivative. However, we have the additional restriction that the denominator be nonzero. This occurs at the point where π‘₯=9. Therefore, the domain of the derivative does not contain this point. Therefore, written in interval notation, the domain of 𝑔(π‘₯) is (βˆ’βˆž,9).

It is interesting to note that the derivative might only be defined on a subset of the domain of the original function. When we look at the differentiability of a function, we will explore this in more detail.

Example 5: Evaluating Limits Using Derivatives

Evaluate limο‚β†’οŠ¦π‘“(β„Ž+4)βˆ’π‘“(β„Žβˆ’2)+𝑓(βˆ’2)βˆ’π‘“(4)β„Ž.

Answer

When asked to evaluate a limit, often we need to relate it to other things we know. Here, we can see that it somehow looks similar to the definition of the derivative of 𝑓: 𝑓(π‘₯)=𝑓(π‘₯+β„Ž)βˆ’π‘“(π‘₯)β„Ž.οŽ˜ο‚β†’οŠ¦lim

We can therefore, try to rearrange the expression to see whether we can isolate the definition of the derivative of 𝑓 at a particular point. Therefore, limlimο‚β†’οŠ¦ο‚β†’οŠ¦π‘“(β„Ž+4)βˆ’π‘“(β„Žβˆ’2)+𝑓(βˆ’2)βˆ’π‘“(4)β„Ž=(𝑓(β„Ž+4)βˆ’π‘“(4))(βˆ’π‘“(β„Žβˆ’2)+𝑓(βˆ’2))β„Ž.

Since β„Ž+4=4+β„Ž, we can see that the first part of this expression is the derivative 𝑓(4). Hence, using the rules of limits, we have limlimlimlimο‚β†’οŠ¦ο‚β†’οŠ¦ο‚β†’οŠ¦οŽ˜ο‚β†’οŠ¦π‘“(β„Ž+4)βˆ’π‘“(β„Žβˆ’2)+𝑓(βˆ’2)βˆ’π‘“(4)β„Ž=(𝑓(4+β„Ž)βˆ’π‘“(4))β„Žβˆ’π‘“(β„Žβˆ’2)βˆ’π‘“(βˆ’2)β„Ž=𝑓(4)βˆ’π‘“(β„Žβˆ’2)βˆ’π‘“(βˆ’2)β„Ž.

Similarly, since β„Žβˆ’2=βˆ’2+β„Ž, we can recognize this second expression as the derivative 𝑓(βˆ’2). Therefore, limο‚β†’οŠ¦οŽ˜οŽ˜π‘“(β„Ž+4)βˆ’π‘“(β„Žβˆ’2)+𝑓(βˆ’2)βˆ’π‘“(4)β„Ž=𝑓(4)βˆ’π‘“(βˆ’2).

When we gave the intuitive idea of the derivative, we relied on the fact that as we zoom in on a curve it increasingly resembles a straight line. However, this is not the case for all functions. In fact, there are many functions where this is not the case. In the next example, we will consider one such function.

Example 6: Differentiability of a Function

Consider the function 𝑓(π‘₯)=|π‘₯|.

  1. Find limο‚β†’οŠ¦οŽ©π‘“(β„Ž)β„Ž.
  2. Find limο‚β†’οŠ¦οŽͺ𝑓(β„Ž)β„Ž.
  3. What can you conclude about the derivative of 𝑓(π‘₯) at π‘₯=0?

Answer

Part 1

We are considering the limit from the right as β„Ž tends to zero. Since |π‘₯|=π‘₯ for all π‘₯β‰₯0, we have limlimlimο‚β†’οŠ¦ο‚β†’οŠ¦ο‚β†’οŠ¦οŽ©οŽ©οŽ©π‘“(β„Ž)β„Ž=β„Žβ„Ž=1=1.

Part 2

Similarly, we can consider the limit from the left as β„Ž tends to zero. Since |π‘₯|=βˆ’π‘₯ for all π‘₯<0, we have limlimlimο‚β†’οŠ¦ο‚β†’οŠ¦ο‚β†’οŠ¦οŽͺοŽͺοŽͺ𝑓(β„Ž)β„Ž=βˆ’β„Žβ„Ž=βˆ’1=βˆ’1.

Part 3

Since the right and left limits are unequal, the limit limlimο‚β†’οŠ¦ο‚β†’οŠ¦π‘“(β„Ž)β„Ž=𝑓(β„Ž)βˆ’π‘“(0)β„Ž does not exist. We can, therefore, conclude that the derivative of 𝑓 does not exist at π‘₯=0.

Key Points

  • The derivative of a function is defined as limο‚β†’οŠ¦π‘“(π‘₯+β„Ž)βˆ’π‘“(π‘₯)β„Ž. An alternative but equivalent definition of the derivative is limο—β†’ο—οŠ§οŠ¦οŠ§οŠ¦οŽ οŽŸπ‘“(π‘₯)βˆ’π‘“(π‘₯)π‘₯βˆ’π‘₯. There are two common ways we denote derivatives: Leibniz’s notation and prime notation (sometimes referred to as Lagrange’s notation). For a function 𝑦=𝑓(π‘₯), Leibniz’s notation for the derivative is written using the infinitesimals d𝑦 and dπ‘₯ as dd𝑦π‘₯, which we read as β€œthe derivative of 𝑦 with respect to π‘₯” or β€œd𝑦 by dπ‘₯.”
    Using prime notation, the derivative of 𝑓(π‘₯) with respect to π‘₯ is denoted 𝑓(π‘₯) which we read as β€œπ‘“ prime of π‘₯.”
  • The derivative defines a function which is equal to the slope of the tangent at each point on the curve.
  • A continuous function is not necessarily differentiable at all points in the domain of the function.

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