How to find the difference quotient of a function

For a function f(x), you can find its difference quotient by using the formula

Difference quotient = [  f(x + h) - f(x) ] / h

The difference quotient is a slope of f(x). Remember that the difference quotient of a function can be a number, or an algebraic expression containing 'h'. Example 1 below has a difference quotient of a number, and example 2 has a difference quotient of an algebraic expression containing 'h'.


The steps to calculate difference quotient are as follows:

• Calculate f(x + h)
• Substitute f(x) and f(x + h) in the formula above
• Simplify

Example 1: Calculate the difference quotient of f(x) = x + 1

Step 1: Find f(x + h)

f(x + h) = (x + h) + 1

= x + h + 1

Step 2: Substitute f(x) and f(x + h) in the formula

Difference quotient = [ (x + h + 1) - (x + 1) ] / h

Step 3: Simplify

= h / h

= 1

So the difference quotient of f(x) = x + 1 is 1

Example 2: Find the difference quotient of f(x) = 2x^2 + x + 1

Step 1: Find f(x + h)

f(x + h) = 2(x + h)^2 + (x + h) + 1

= 2x^2 + 2h^2 + 4xh + x + h +1

 Step 2: Substitute f(x) and f(x + h) in the above formula

Difference quotient = [ (2x^2 + 2h^2 + 4xh + x + h +1) - (2x^2 + x + 1) ] / h

Step 3: Simplify

 = [ 2h^2 + 4xh + h ] / h

= 2h + 4x + 1

 Difference quotient of f(x) = 2x^2 + x + 1 is 2h + 4x + 1