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.

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}

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}


More on
differentiation and derivatives

privacy policy