Difference Quotient

What is the difference quotient in calculus? We start with the definition and then we calculate the difference quotient for different functions.




Web www.analyzemath.com

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 = [ f (x + h) f(x) ] / [ (x + h) x ]


Simplify the denominator to obtain

m = [ f (x + h) f(x) ] / h


This slope is very important in calculus where it is used to define the derivative of function f. 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

    [ f (x + h) f(x) ] / h = [ 2(x + h) + 5 (2 x + 5) ] / h

  • We simplify the above expression.

    = 2 h / h = 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

    [ f (x + h) f(x) ] / h =

    [ 2(x + h) 2 + (x + h) - 2 - ( 2x 2 + x - 2 ) ] / h

  • We expand the expressions in the numerator and group like terms.

    = [ 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

    [ f (x + h) f(x) ] / h =

    [ 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.

    [ f (x + h) f(x) ] / h = 2 cos [ (2 x + h)/2 ] sin (h/2) / h



More on differentiation and derivatives

Find the Inverse Functions - Online Calculator
Free Online Graph Plotter for All Devices
Home Page -- HTML5 Math Applets for Mobile Learning -- Math Formulas for Mobile Learning -- Algebra Questions -- Math Worksheets -- Free Compass Math tests Practice
Free Practice for SAT, ACT Math tests -- GRE practice -- GMAT practice Precalculus Tutorials -- Precalculus Questions and Problems -- Precalculus Applets -- Equations, Systems and Inequalities -- Online Calculators -- Graphing -- Trigonometry -- Trigonometry Worsheets -- Geometry Tutorials -- Geometry Calculators -- Geometry Worksheets -- Calculus Tutorials -- Calculus Questions -- Calculus Worksheets -- Applied Math -- Antennas -- Math Software -- Elementary Statistics High School Math -- Middle School Math -- Primary Math
Math Videos From Analyzemath
Author - e-mail


Updated: 2 April 2013

Copyright © 2003 - 2014 - All rights reserved