Chapter 7 Differential Calculus

Section 7.1 Derivative of a Function

7.1.3 Derivative

Notation of the Derivative 7.1.3
In mathematics, sciences and engineering, different but equivalent notations for derivatives are used:

f'( x0 )= df dx ( x0 )= d dx f( x0 ).

These different notations all denote the derivative of the function f at the point x0 .

If the derivative is to be calculated using the difference quotient f(x)-f( x0 ) x- x0 , then it is often convenient to rewrite the difference quotient in another way. Denoting the difference of x and x0 by h:=x- x0 (see figure below),

the difference quotient can be rewritten as

f(x)-f( x0 ) x- x0 = f( x0 +h)-f( x0 ) h ,

where x= x0 +h. There is no statement about whether x has to be greater or less than x0 . Hence, the quantity h can take positive or negative values. To determine the derivative of the function f, the limit for h0 has to be calculated:

f'( x0 )= limx x0 f(x)-f( x0 ) x- x0 = limh0 f( x0 +h)-f( x0 ) h .

If this limit exists for all points x0 in a function's domain, then the function is said to be differentiable (everywhere). Many of the common functions are differentiable. However, a simple example of a function that is not differentiable everywhere is the absolute value function f: with xf(x):=|x|.
Example 7.1.4
The absolute value function (see Module 6, Section 6.2.5) is not differentiable at the point x0 =0. The difference quotient of f at the point x0 =0 is:

f(0+h)-f(0) h = |h|-|0| h = |h| h .

Since h can be greater or less than 0, two cases are to be distinguished: For h>0, we have |h| h = h h =1, and for h<0, we have |h| h = -h h =-1. In these two cases, the limiting process, i.e. h approaching 0, results in two different values ( 1 and -1). Thus, the limit of the difference quotient at the point x0 =0 does not exist. Hence, the absolute value function is not differentiable at the point x0 =0.
The graph changes its direction at the point (0;0) abruptly: Casually speaking, one says that the graph of the function has a kink at the point (0;0).

Likewise, if a function has a jump at a certain point, a unique tangent line to the graph at this point does not exist and thus, the function has no derivative at this point.