Derivative of Simple Functions

Bridge Course, August 2010

July 28, 2010

Intuition: The derivative of a function f(x) assigns measures the “slope” (i.e. how quickly the value of f(x) changes as the value of x changes) of the function at the point x. It is denoted , or just f′(x).

Figure 1: f(x) = e^x with a tangent to f at x = 2.5

Definition: Consider a function f = f(x) defined over real-line. Then f′(x) is defined as 

                                               .

Examples: Derivative computation

Consider a few examples of derivative computation:

When limits exist                                                                                                                    Maxima CODE

1. Consider f(x) = x. Then                                                                                                          diff(x,x,1)

    Hence the slope of the function is 1 everywhere.

2. Consider f(x) = 2x. Then .                                                            diff(2*x,x,1)

3. Consider f(x) = x2. Then . 

Hence the slope of the function changes with x – in particular, the larger the magnitude of x, the larger the slope.                                                                                                                                                     diff(x∧2,x,1)