Let \( f(x) \) be a function and point \( A(x,f(x))\) and \( B(x+h,f(x+h)) \) on the graph of \( f \) as shown below.

The difference quotient of \( f(x) \) is defined by:
\[ m = \dfrac{f(x+h)-f(x)}{h} \]
which is the slope of the secant line through the points \( A \) and \( B \).
The limit as \( h \) approaches zero of the difference quotient, defined above, gives the important concept of the derivative of a function.

Use of the Difference Quotient Calculator

1 - Enter and edit function $f(x)$ and click "Enter Function" then check what you have entered.
Note that the five operators used are: + (plus) , - (minus), / (division) , ^ (power) and * (multiplication). (example: f(x) = x^3 + 1/x.(more notes on editing functions are located below)
2 - Click "Calculate Quotient".
3 - Note that the final expression of the difference quotient is simplified for polynomial and rational functions.
4 - The use of the present calculator and the definition of the derivative will help fully learn how to calculate derivative of a function using its definition.

Notes: In editing functions, use the following:
1 - The five operators used are: + (plus) , - (minus), / (division) , ^ (power) and * (multiplication). (example: f(x) = x^2 + 1/x + log(x) )
2 - The function square root function is written as (sqrt). (example: sqrt(x^2-1) for \( \sqrt {x^2 - 1} \) )
3 - The exponential function is written as exp(x). (Example: exp(x+2) for \( e^{x+2} \) )
4 - The log base e function is written as log(x). (Example: log(x^2-2) for \( \ln(x^2 - 2 \) )
Here are some examples of functions that you may copy and paste to practice:
x^2 3 x^2 + 2x 1/x 1 / (x -2) (x-2)/(x+3)
sin(2x+1) exp(x -2) tan(x) (x-1)/(x+3)^2