What Is The Difference Quotient And How To Calculate It?

What Is The Difference Quotient And How To Calculate It?

As part of understanding the definition of derivatives, it is important first to understand the concept of the difference quotient and its formula.

Unlike a percentage or an average, the word “derivative” simply refers to the amount of change in a function. At any moment, the rate of change at any given point is the instantaneous rate of change. A tangent line is used for the measurement of this value. Using this, we can obtain the best possible linear approximate prognosis at a single point by measuring its slope.

However, for the sake of simplicity, we don’t intend to discuss derivatives specifically in this article. Rather than calculating the derivatives of functions, we will take a look at the difference quotient, which can serve as a starting point towards calculating them.

As a consequence of the difference quotient, we can derive the slope of secant lines. As with tangent lines, secant lines carry over at least two points on a function, but they are nearly the same as tangent lines.

Difference Quotient Formula:


  • f(x) and f(a) are both same.
  • h = Difference between x-values

Although our theory is derived from a line, it still holds regardless of the type of function. Hence, this quotient difference formula is the only quotient difference formula you need to know!

It is common to find questions regarding how to compute a difference quotient for a given function or how to simplify    for each function.

As soon as you become acquainted with it, it’s easy to set up and use. It can be extremely challenging to simplify rational functions and radical functions, but with practice, it becomes easier and easier.

To practice, look through examples. But before we do that, let’s make sure that you’re comfortable with some basics of function operations by watching our videos on function notation, dividing functions, composite functions, and slope equations. By mastering all of these techniques, you will be able to master the difference quotient much more easily.

As the difference quotient can be formulated in a linear manner, it simply requires that the right terms be substituted for the formula.  The best way to simplify the process is to start by taking a good look at the given equation. For instance:

f(x) = 2x^{2} + 5x + 4f(x)=2x2+5x+4

f(x) = 3(x-4x)^{2} – 3x + 4(-8)f(x)=3(x−4x)2−3x+4(−8)

Therefore, when the difference quotient formula is applied to terms, the first equation is far easier than the second. As a result, the second equation involves a greater number of operations, and the inclusion of “h” increases the chances of errors. As we examine some example problems, this becomes clearer.

difference quotient formula
difference quotient formula

Derive Difference Quotient For Linear Functions:

When setting up a difference quotient, linear functions are easiest. When it comes to work of this nature, the fewer variables and the lower the degree the easier it is!

Example No. 1:

Find out the difference quotient of equation f(x) = 2x+5f(x) =2x+5

This is a 2-Step calculation and will show you both steps one-by-one in detail.

Step 1

Here’s the difference quotient formula again.

Let us now find f(x+h) by substituting values of (x + h) into f(x).

f(x+h) = 2(x+h) + 5f(x+h) = 2(x+h) + 5

Now, we can put f(x + h) and f(x) into the formula.


Step 2: To Simplify the Equation


Solving the Equation:

=  = 2

Difference quotient of f(x) = 2x+5 is 2

Difference Quotient For Polynomial Functions

The difference quotient of polynomial functions also presents similar challenges. A polynomial gets more difficult as the degree (the highest exponent) increases. The most common difference quotient problems will be those involving polynomial functions with a degree of 2.

Additionally, we will examine a polynomial function of a higher degree by constructing a difference quotient for a cubic function.

Example No.2:

Let us find out the difference quotient of f(x) = x2 + 4x – 6

Step 1

As with the difference quotient, we simply set f(a) equal to f(x) and adjust the calculations as needed to accommodate “h”. Keep an eye out for the things in the parentheses; as if you do not, you are more likely to make mistakes! You should ensure you address these kinds of problems when the function is expanded out fully. Having this function factored in would have made the quotient setup more difficult.

= (x + h)2 + 4(x + h) – 6 – (x2 + 4x – 6)

Step 2

= (x + h)2 + 4(x + h) – 6 – (x2 + 4x – 6)

= x2 +2xh + h2 + 4x + 4h – 6 – x2 – 4x + 6

= 2xh + h2 + 4h  

= h(2x + h + 4)  

= 2x + h + 4

Final Thoughts

There you have it, it’s as simple as that! This article has covered the basics of calculating difference quotient and gives you a few methods that you can use. But your educational journey is not over yet! If you want to be confident in your understanding of this topic, make sure that you practice even more complex functions.

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