1.4 Higher-order derivatives

In Exercise 2 you saw that the function has derivative , a result that you can also obtain using the Reciprocal Rule. If you now apply the Reciprocal Rule to the derivative , then you obtain a function

In general, for a differentiable function , the function  is called the second derivative of , and is denoted by . Continued differentiation gives the so-called higher-order derivatives of . These are denoted by , and the values , are called the higher-order derivatives of at .

Since the dashes in this notation can be rather cumbersome, we often indicate the order of the derivative by a number in brackets. Thus mean the same as , respectively. Here the brackets in are needed to avoid confusion with the fourth power of .

When we wish to discuss a derivative of general order, we will refer to the th derivative of . It is often possible to find a formula for the th derivative in terms of . For example, if , then

so the th derivative is given by

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

One interesting feature about this formula is that the domain remains the same, no matter how often the function is differentiated. This is a special case of a much more general result which states that: a function that is analytic on a region has derivatives of all orders on . 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.