# Difference Quotient

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

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 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 $\dfrac{f (x + h) – f(x)}{h}$ This slope is very important in calculus where it is used to define the derivative of function $f$. 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. $= 2 h / h = 2$ 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 $f(x) = 2 x^2 + x - 2$ 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 $\dfrac{f (x + h) – f(x)}{h}=$ $\dfrac{2(x + h)^2 + (x + h) - 2 - ( 2x^2 + x - 2 )}{h}$ 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. $\sin (x + h) - \sin x = 2 \cos [ \dfrac{2 x + h}{2}] \sin (\dfrac{h}{2})$ We substitute the above expression for $\sin (x + h) - \sin x$ in the difference quotient above to obtain. $\dfrac{f (x + h) – f(x)}{h}= \dfrac{2 \cos [ \dfrac{2 x + h}{2}] \sin (\dfrac{h}{2})}{h}$ More on differentiation and derivatives

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Updated: 2 April 2013