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 QuotientLet 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 1Find the difference quotient of function f defined by
Solution to Example 1
Example 2Find the difference quotient of the following function
Solution to Example 2
Example 3Find the difference quotient of function f given by
Solution to Example 3
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