✅ Difference Quotient Formula ⭐️⭐️⭐️⭐️⭐

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Difference Quotient Formula

The difference quotient formula is a part of the definition of the derivative of a function. By taking the limit as the variable h tends to 0 to the difference quotient of a function, we get the derivative of the function. Let us learn the difference quotient formula along with its derivation and examples.

What Is the Difference Quotient Formula?

By seeing the name, “difference quotient formula”, are you able to recollect something? The words “difference” and “quotient” are giving the feel of the slope formula. Yes, the difference quotient formula gives the slope of a secant line that is drawn to a curve. What is a secant line? A secant line of a curve is a line that passes through any two points of the curve. Let us consider a curve y = f(x) and a secant line that passes through two points of the curve (x, f(x)) and (x + h, f(x + h)). Then the difference quotient of the function f(x) is shown below.

Difference Quotient Formula

The difference quotient formula of a function y = f(x) is,

[ f(x + h) – f(x) ] / h

where

  • f (x + h) is obtained by replacing x by x + h in f(x)
  • f(x) is the actual function

Difference Quotient Formula Derivation

Let us consider a function y = f(x) and let a secant line passes through two points of the curve (x, f(x)) and (x + h, f(x + h)).  Thus using the slope formula, the slope of the secant line is,

[ f(x + h) – f(x) ] / [ (x + h) – x] = [ f(x + h) – f(x) ] / h [Since the slope of any straight line = change in y/ change in x.)

This is nothing but the difference quotient formula.

Note: As h → 0, the secant of y = f(x) becomes a tangent to the curve y = f(x). Thus, as h → 0, the difference quotient gives the slope of the tangent and hence it gives the derivative of y = f(x). i.e.,

f ‘ (x) = lim h→0h→0 [ f(x + h) – f(x) ] / h

Examples Using Difference Quotient Formula

Example 1: Find the difference quotient of the function f(x) = 3x – 5.

Solution:

Using the difference quotient formula, 

Difference quotient of f(x) 

= [ f(x + h) – f(x) ] / h

= [ (3(x + h) – 5) – (3x – 5) ] / h

= [ 3x + 3h – 5 – 3x + 5 ] / h

= [ 3h ] / h

= 3

Answer: The difference quotient of f(x) is 3.  

Example 2 : Find the derivative of f(x) = 2x2 – 3 by applying the limit as h → 0 to the difference quotient formula.

Solution:

The difference quotient of f(x) 

= [ f(x + h) – f(x) ] / h

= [ (2(x + h)2 – 3) – (2x2 – 3) ] / h

= [ (2 (x2 + 2xh + h2) – 3) – 2x2 + 3 ] / h

= [ 2x2 + 4xh + 2h2 – 2x2 + 3 ] / h

= [ 4xh + 2h2 ] / h

= [ h (4x + 2h) ] / h

= 4x + 2h

By applying the limit as h → 0, we get the derivative f ‘ (x).

f ‘(x) = 4x + 2(0) = 4x.

Answer: f ‘ (x) = 4x.

Example 3: Find the difference quotient of the function f(x) = ln x.

Solution:

Using the difference quotient formula, the difference quotient of f(x) is,

[ f(x + h) – f(x) ] / h

= [ ln (x + h) – ln x ] / h

=  ln [ (x + h) / x ] / h (because by quotient property of logarithms, ln m – ln n = ln (m / n))

Answer: The difference quotient of f(x) is, ln [ (x + h) / x ] / h.

FAQs on Difference Quotient Formula

What Is Difference Quotient Formula?

The difference quotient formula is nothing but the slope of a secant line formula. The difference quotient of a function y = f(x) is given by [ f(x + h) – f(x) ] / h.

How To Derive Difference Quotient Formula?

Since the difference quotient is nothing but the slope of a secant line, we use the slope formula to derive the difference quotient formula. The slope of the line joining (x, f(x)) and (x + h, f(x + h)) by slope formula is, [ f(x + h) – f(x) ] / [ (x + h) – x] = [ f(x + h) – f(x) ] / h. This is the difference quotient formula.

What Are the Applications of the Difference Quotient Formula?

The difference quotient formula is mainly used to find the derivative. i.e., the limit of the difference quotient as h → 0 gives the derivative of the function. i.e., f ‘ (x) = lim  [ f(x + h) – f(x) ] / h

How To Use the Difference Quotient Formula To Find the Derivative?

The limit of the difference quotient of a function f(x) as h → 0 is nothing but the derivative of the function. i.e., the derivative of the function f(x) is,  f ‘ (x) = lim  [ f(x + h) – f(x) ] / h.

Difference quotient: applications of functions – Relations and Functions

Difference Quotient

What is the Difference Quotient?

Before we define the difference quotient and the difference quotient formula, it is important to first understand the definition of derivatives.

In the most basic sense, a derivative is a measure of a function’s rate of change. That is, the instantaneous rate of change at a given point. In order to measure this value, we use what is called a tangent line, and measure its slope to given the best possible linear approximation at a single point.

In this article, however, we won’t be looking at derivatives in particular. Instead, we will be looking at the difference quotient, which is a stepping stone to calculating derivatives of functions. The difference quotient allows us to compute the slope of secant lines. A secant line is nearly the same as a tangent line, but it instead goes through at least two points on a function.

In order to understand what the difference quotient is, it helps to visualize two points on a line like the following image:

Visualize two points on a line to understand difference quotient

If we recall the definition of slope, we can use this definition to find the “secant line” going through two points. Furthermore, if we use the variable “h” to represent the difference between the two x-values, we start to see a formula developing:

Develop a formula using definition of slope, secant line and variable h

Finally, with some cancelling of terms, we can arrive at the very definition of the difference quotient.

Cancel terms to get the definition of difference quotient

Therefore, the formula for the difference quotient is:

much easier than the second. This is because the second equation involves a lot more operations, and the addition of “h” into that mix makes errors much more likely. This will become more apparent as we get into some example problems.

Set Up a Difference Quotient for Linear Functions:

As it often goes, setting up the difference quotient is easiest for linear functions. For this kind of work, the less variables and the lower the degree, the easier it is!

Example 1:

Again, in order to use the difference quotient, all we need to do is set f(a) equal to f(x), and make the necessary modifications to accommodate “h”. Also remember it is very important to keep track of what is in the parentheses, as this is where it is most common to make mistakes! As an important side note, make sure you tackle these kinds of problem when the function is expanded out fully! If this function had been in factored form, it would have been much harder to set up the quotient.

Just like with the quadratic function in the previous example, in order to use the difference quotient, all we need to do is set f(a) equal to f(x), and make the necessary modifications to accommodate “h”. Also remember it is very important to keep track of what is in the parentheses, as this is where it is most common to make mistakes! This cannot be more important as we start dealing with more difficult equations and functions. Finally, just like the last example, make sure you set up the difference quotient for the function in its fully expanded form, i.e. not factored!

Set up difference quotient

STEP 2: Simplify

Once we have set up the difference quotient, all we need to do is expand whatever is in the brackets, collect like terms, and simplify! This is much more difficult compared to the quadratic example, as we have many more terms! Be sure to double and triple check your work. It is very easy to lose track of terms in these kinds of questions.

Simplify

Once again, even though we have a rational function, which is a bit more tricky, the technique is still the same. In order to use the difference quotient, all we need to do is set f(a) equal to f(x), and make the necessary modifications to accommodate “h”. Since we are using a rational function, you need to pay extra attention to detail to make sure everything is set up correctly.

In this example, a slightly difference method is shown below. Instead of simply trying to input all values into the formula for the difference quotient, the first step is to instead find f(x+h) first. Once we know f(x+h), can can sub that into the difference quotient equation.

Set up difference quotient

STEP 2: Simplify

After the slightly different first step, the rest is the same. Keeping in mind the careful attention we need to have, all we need to do is expand whatever is in the brackets, collect like terms, and simplify! Then, we can easily find our final answer.

Simplify

Set Up a Difference Quotient for Radical Functions:

Last, but certainly not least, is finding the difference quotient for radical functions. By their very nature, radical functions are a bit more difficult than anything else. But, if we stay we the same technique we’ve been using since linear functions, nothing is all that different!

Example 6:

Difference Quotient – Definition, Formula, & Examples

When you hear of the difference quotient, what are the math concepts that you have in mind? What’s an algebra concept that involves differences and quotients? Well, if you guessed slope, you’re actually close to the definition of the difference quotient.

The difference quotient measures the slope of a secant line passing through the curve of f(x).

There are two important keywords here that we need to understand: slope and secant line.

A linear function’s slope represents the measure of its “rise over run”. What does this mean mathematically?

Now, what are secant lines? They are the lines that pass through a figure or a graph at exactly two points.

The graph shown above is an example of a secant line passing through a curve. As you can see, the secant line passes through two points along the curve.

Now that we’ve reviewed these important concepts, it’s that we learn more about difference quotients.

What is the difference quotient?

Understanding the difference quotient formula

So, how do we apply this definition and calculate for the difference quotient of a function given a secant line? We use our knowledge of slope to establish the formula for the difference quotient.

How to simplify difference quotient?

of a function. This example will show how, with the right technique, we can actually manipulate the expressions faster. What are we waiting for? Let’s begin!

How to Solve the Difference Quotient

Steps to Solve

The difference quotient is an important part of mathematics, especially calculus. Given a function f(x), and two input values, x and x + h (where h is the distance between x and x + h), the difference quotient is the quotient of the difference of the function values, f(x + h) – f(x), and the difference of the input values, (x + h) – x.

Continuing on, we can simplify the denominator of this because the x cancels out, which you can see play out here.

We see that the difference quotient is as follows:

(f(x + h) – f(x)) / h

Great! We’ve got a formula for the difference quotient. Now, let’s consider the steps involved in solving this difference quotient. Any time we are given a formula, solving the formula is just a matter of finding the values of the variables and expressions involved in the formula, plugging them into the formula, and then simplifying. Therefore, the following steps are used to solve the difference quotient for a function, f(x).

  1. Plug x + h into the function f and simplify to find f(x + h).
  2. Now that you have f(x + h), find f(x + h) – f(x) by plugging in f(x + h) and f(x) and simplifying.
  3. Plug your result from step 2 in for the numerator in the difference quotient and simplify.

Okay, three steps. We can handle this! Let’s try an example. Suppose we want to find the difference quotient for the function

g(x) = x2 + 3

As you can see here, we start by plugging the expression x + h into g, and simplifying.

As we can see, we get that g(x + h) = x2 + 2xh + h2 + 3. Next, we plug g(x + h) and g(x) into g(x + h) – g(x) and simplify.

As you can see, we end up with 2xh + h2. So far, so good! All we have left to do is plug this in for the numerator in the difference quotient and simplify.

We see that given the function g(x) = x2 + 3 and two input values of x and x + h, the difference quotient is 2x + h, where h is the difference between the two input values. That’s not so bad!

Practice Questions to Solve a Difference Quotient (Show All Your Work)

1. Find the difference quotient for the function f(x) = 1/x (reciprocal function).

2. Find the difference quotient for the function g(x) = x^3 (cube function).

Answers (To Check Your Work)

1. The difference quotient for f(x) can be expressed as follows:

[f(x + h) – f(x) ] / h

= [1/(x + h) – 1/x ] / h

Using LCD = x(x + h) gives us the difference quotient as:

= [x – (x + h)] / [hx(x + h)]

= [x – x – h] / [hx(x + h)]

= -h / [hx(x + h)]

= -1 / [x(x + h)].

2. The difference quotient for g(x) can be expressed as follows:

[g(x + h) – g(x) ] / h

= [(x + h)^3 – x^3 ] / h

Using (x + h)^3 = x^3 + 3 x^2 h + 3 x h^2 + h^3 leads to the difference quotient as:

= [x^3 + 3 x^2 h + 3 x h^2 + h^3 – x^3] / h

= [3x^2 h + 3 x h^2 + h^3] / h

Pulling out an h from the numerator above gives us the difference quotient as:

= h[3x^2 + 3x h + h^2] / h

= 3x^2 + 3x h + h^2.

Note: Passing the limit as h goes to zero gives us the first derivative of the function at x for both 1 and 2 above.

Difference Quotient

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