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### 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