# 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.

## 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}

This slope is very important 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. It is called the
difference quotient. 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 graph of 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

### 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 sin (x + h) - sin x 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}