#### Difference Quotient: Formula, Derivation, and Examples

Definition of the derivative of a function includes the difference quotient formula. By taking the limit as the variable h tends to 0 to the difference quotient, we get the function’s derivative. In addition to its derivation, let’s understand the difference quotient formula.

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

Derived functions are measures of the rate at which a function changes. In other words, it’s the rate at which things are changing at a given moment. We use a tangent line to measure this value and calculate its slope to provide the best possible linear approximation at a single point.

Despite that, we won’t be focusing on derivatives specifically in this article. We will instead focus on the difference quotient, which serves as a stepping stone to calculating derivatives. Using the difference quotient, we can calculate the slope of secant lines. In comparison to a tangent line, a secant line passes through at least two points on a function.

**Note**: f(x) and f(a) are the same thing and are just different notations for writing the formula. In either case, “h” represents the difference between the two values.

In spite of the fact that we developed this theory from a graph, it holds for all functions. Hence, you need only memorize this formula for difference quotients!

Different types of questions can be asked when using the difference quotient. Typically, the question asks either to set up a difference quotient or to simplify or frac [f (a + h) -f (a)] [h] hf (a + h) – f (a) for the given function. Luckily, it is straightforward to set up and use once you are familiar with it. It isn’t easy to simplify with rational and radical functions, but it is still more than manageable with lots of practice!

You can always practice this by looking at some examples. We’ll introduce function notation and dividing functions next, and then we’ll tackle composite functions and the slope equation. As we are skilled in all of these methods, using the difference quotient will be easier for us.

**What Is the Difference Quotient Formula? **

When you think of the “difference quotient formula” do you remember anything? There is an underlying sense of slope formula in the words “difference” and “quotient”. It is true, a secant line drawn to a curve gives its slope by using the difference quotient formula.

What is a secant line? Secant lines are lines that pass through two points in a curve. There are two points on the curve (x, f(x)) and (x + h, f(x + h)) on which a secant line passes through. As a result, the difference quotient is presented below.

**Difference Quotient Formula **

A function y = f(x) has a difference quotient formula as follows:

[ f(x + h) – f(x) ] / h

Where

- By replacing x by x + h in f(x), we obtain f (x + h).
- f(x) is the actual function

**Difference Quotient Formula Derivation**

The secant line passes through two points of the curve (x, f(x)) and (x + h, f(x + h)). Therefore, using the slope formula, the secant line slope 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 more than the difference quotient formula.

In the case of h = 0, y = f(x) becomes a tangent as h * 0. 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 \(_{h → 0}\) [ f(x + h) – f(x) ] / h

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Next, we will examine the applications of the difference quotient formula.

**Examples Using the Difference Quotient Formula**

**Here’s an example: 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

**F(x) has a difference quotient of 3.**

**Using the difference quotient formula, find the derivative of f(x) = 2×2 – 3 by applying the limit as h * 0.**

**Solution:**

The difference quotient of f(x)

= [ f(x + h) – f(x) ] / h

= [ (2(x + h)2 – 3) – (2×2 – 3) ] / h

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

= [ 2×2 + 4xh + 2h2 – 2×2 + 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: **Calculate the difference quotient of the function f(x) = ln x.

**Solution:**

According to the difference quotient formula, the difference quotient of f(x) is:

[ f(x + h) – f(x) ] / h

= [ ln (x + h) – ln x ] / h

The logarithm property says that ln [m + n] = ln (m / n) (because of the quotient property of logarithms)

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

**FAQs**

**What Is the Difference Quotient Formula?**

Basically, the difference quotient formula is nothing more than a secant line’s slope. The difference quotient of a function y = f(x) is equal to [ f(x + h) – f(x) ] / h.

**How to Derive Difference Quotient Formula?**

As the difference quotient is nothing but the slope of a secant line, we can derive the difference quotient formula using the slope 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.

**Difference Quotient Formula: What Are the Applications?**

Differential equations are often solved using the difference quotient formula. By definition, the derivative of a function can be obtained by dividing h by zero. i.e., f ‘ (x) = lim\(_{h → 0}\) [ f(x + h) – f(x) ] / h

**How Do You Find the Derivative of a Difference Quotient Formula?**

The limit of the difference quotient of a function f(x) is nothing more than the derivative of the function. Specifically, f'(x) = lim/(_[h * 0]/) [ f(x + h) – f(x) ] / h is the derivative of f(x).