## 6.5. Differentiable Functions

### Examples 6.5.4(a):

**R**..

Let us try to use the original definition of differentiability: we would like to call the function f(X) = f(x, y, z) differentiable in a point C = (a, b, c) if the limit of the difference quotient

- ( f(X) - f(C) ) / (X - C)

as X approaches C exists. The numerator is well-defined, since f(X) and f(C) are real numbers, so that their difference can be computed as usual. For the denominator we can define the difference of X - C = (x, y, z) - (a, b, c) by taking the differences in each component, i.e. X - C = ( x - a, y - b, z - c). However, we have problems defining division. To make sense of the above difference quotient we must know how to define the quotient of a real number and a vector X = (x, y, z).

There is no satisfying definition of such a quotient. Therefore, we can not use this difference quotient in this situation to define 'derivative'.

To avoid this problem, we need a single real number in the denominator as well. Hence, we could look, for example, at the quotient

- ( f(X) - f(C) / (x - a)

where X = (x, y, z) and C = (a, b, c) and take the limit as x
approaches c with all other variables fixed. This concept, somewhat
modified, is known as** partial derivatives**.

Thus, we can use our original definition of differentiability to define partial derivatives, but not to define 'the derivative' of a function of more than one variable.

**Note:** There is one notable exception - the **space of
complex numbers**. Recall that a complex number z = x + i y
consists of a real and an imaginary part, where i is the square
root of -1. Thus, a complex number z = x + i y can be identified
with the tuple z = (x, y). And there is a way in two dimensional
space to define division (and multiplication) in a meaningful
way (how ?). Therefore, if we identify two-dimensional real space
with the space of complex numbers, we can use the original definition
of derivative to define the complex derivative of a complex function.