1.4 Higher-order derivatives

In Exercise 2 you saw that the function f of z equals one solidus z has derivative f super prime of z equals negative one solidus z squared, a result that you can also obtain using the Reciprocal Rule. If you now apply the Reciprocal Rule to the derivative f super prime of z equals negative one solidus z squared, then you obtain a function

left parenthesis f super prime right parenthesis super prime times left parenthesis z right parenthesis equals two divided by z cubed times left parenthesis z not equals zero right parenthesis full stop

In general, for a differentiable function f, the function left parenthesis f super prime right parenthesis super prime is called the second derivative of bold-italic f, and is denoted by f super prime prime. Continued differentiation gives the so-called higher-order derivatives of bold-italic f. These are denoted by f super prime prime comma f super prime prime prime comma f super prime prime prime prime comma ellipsis, and the values f super prime prime of alpha comma f super prime prime prime of alpha comma f super prime prime prime prime of alpha comma ellipsis, are called the higher-order derivatives of bold-italic f at bold-italic alpha.

Since the dashes in this notation can be rather cumbersome, we often indicate the order of the derivative by a number in brackets. Thus f super left parenthesis two right parenthesis comma f super left parenthesis three right parenthesis comma f super left parenthesis four right parenthesis comma ellipsis mean the same as f super prime prime comma f super prime prime prime comma f super prime prime prime prime comma ellipsis, respectively. Here the brackets in f super left parenthesis four right parenthesis are needed to avoid confusion with the fourth power of f.

When we wish to discuss a derivative of general order, we will refer to the bold-italic nth derivative bold-italic f super left parenthesis bold-italic n right parenthesis of bold-italic f. It is often possible to find a formula for the nth derivative in terms of n. For example, if f of z equals one solidus z, then

f super prime prime of z equals two divided by z cubed comma f super prime prime prime of z equals negative two multiplication three divided by z super four comma f super left parenthesis four right parenthesis of z equals two multiplication three multiplication four divided by z super five comma ellipsis comma

so the nth derivative is given by

f super left parenthesis n right parenthesis of z equals left parenthesis negative one right parenthesis super n times n factorial divided by z super n plus one full stop

(This can be proved formally by the Principle of Mathematical Induction.)

One interesting feature about this formula is that the domain script cap r equals double-struck cap c minus zero remains the same, no matter how often the function f is differentiated. This is a special case of a much more general result which states that: a function that is analytic on a region script cap r has derivatives of all orders on script cap r. Here we confine our attention to first derivatives, and we continue to do this in the next subsection by giving a geometric interpretation of the first derivative.