# 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. 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