Difference Quotient

What is the difference quotient in calculus ?
We start with the definition and then we calculate the difference quotient for different functions as examples with detailed explanations.
Note that a difference quotient calculator is included and may be used to check results and generate further practice.

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Definition of Difference Quotient

Let \( f \) be a function whose graph is shown below.
graphs of function f with secant line

A and B are points on the graph of \( f\). A line passing trough the two points \( A ( x , f(x)) \) and \( B(x+h , f(x+h)) \) is called a secant line. The slope \( m \) of the secant line may be calculated as follows:
\[ m = \dfrac{f (x + h) - f(x)}{(x + h) - x} \]
Simplify the denominator to obtain
\[ m = \dfrac{f (x + h) - f(x)}{h} \]
The slope \( m \) is called the difference quotient. It is a very important concept in calculus where it is used to define the derivative of function \( f \) which in fact defines the local variation of a function in mathematics.

Examples with Solutions

In the examples below, we calculate and simplify the difference quotients of different functions.

Example 1

Find the difference quotient of function \( f \) defined by \[f(x) = 2x + 5\]

Solution to Example 1

Example 2

Find the difference quotient of the following function
\[ f(x) = 2x^2 + x - 2 \]

Solution to Example 2

Example 3

Find the difference quotient of function \( f \) given by \[ f(x) = \sin x \] and write the result as a product.

Solution to Example 3

More References and links

Difference Quotient Calculator
differentiation and derivatives
Difference quotient