# 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.



## Definition of Difference Quotient

Let $$f$$ be a function whose graph is shown below.

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

• We first need to calculate $$f(x + h)$$.
$$f(x + h) = 2(x + h) + 5$$
• We now substitute $$f(x + h)$$ and $$f(x)$$ in the definition of the difference quotient by their expressions
$$\dfrac{f (x + h) - f(x)}{h} = \dfrac{2(x + h) + 5 - (2 x + 5) }{h}$$
• We simplify the above expression.
$$= \dfrac{2h}{2} = 2$$
• The answer is 2 which also the slope of the line defined by function $$f$$, why?

### Example 2

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

### Solution to Example 2

• We first calculate $$f(x + h)$$.
$$f(x + h) = 2(x + h)^2 + (x + h) - 2$$
• We now substitute $$f(x + h)$$ and $$f(x)$$ in the difference quotient
$$\dfrac{f (x + h) - f(x)}{h} = \dfrac{ 2(x + h)^2 + (x + h) - 2 - ( 2 x^2 + x - 2 )}{h}$$
• We expand the expressions in the numerator and group like terms.
$$= \dfrac{ 4 x h + 2 h^2 + h}{h} = 4 x + 2 h +1$$

### 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

• We first calculate $$f(x + h)$$.
$$f(x + h) = \sin (x + h)$$
• We now substitute $$f(x + h)$$ and $$f(x)$$ in the difference quotient
$$\dfrac{f (x + h) - f(x)}{h} = \dfrac{ \sin (x + h) - \sin x}{h}$$
• We use the trigonometric formula that transform a difference $$\quad \sin (x + h) - \sin x \quad$$ into a product.
$$\sin (x + h) - \sin x = 2 \cos [ (2 x + h)/2 ] \sin (h/2)$$
• We substitute the above expression for $$sin (x + h) - sin x$$ in the difference quotient above to obtain.
$$\dfrac{f (x + h) - f(x)}{h} = \dfrac{ 2 \cos [ (2 x + h)/2 ] \sin (h/2)}{h}$$

### More References and links

Difference Quotient Calculator
differentiation and derivatives
Difference quotient