Derivatives of Higher Order

 Let us try the effect of repeating several times over the operation of differentiating a function (see the concept of a function). Begin with a concrete case.

Let .

There is a certain notation, with which we are already acquainted, used by some writers, that is very convenient. This is to employ the general symbol  for any function of . Here the symbol  is read as “function of,” without saying what particular function is meant. So the statement merely tells us that is a function of , it may be or , or or any other complicated function of .

The corresponding symbol for the derivative is , which is simpler to write than . This is called the derivative of with respect to , the derivative of the function , or simply the derivative function. Instead of or , we sometimes simply write .

Suppose we differentiate over again, we shall get the second derivative of or the second derivative of with respect to , which is denoted by  or ; and so on.

Now let us generalize.

Let .

In general, after differentiating the original function times, we obtain the -th derivative of or with respect to , also known as the derivative of order . When the differentiation order reaches four or more, rather than repeatedly using accents (also known as primes), a more streamlined approach is often adopted. The order of differentiation is denoted using parentheses, with the derivative order presented as a superscript to or . This notation is not only clearer but also helps reduce the risk of miscounting the number of primes. For instance, we often write or instead of and .

There is another way of indicating successive differentiations. For, and this is more conveniently written as , or more usually . Similarly, we may write as the result of differentiating three times, .

How to Read the Symbols for Derivatives

Examples

Now let us try .

In a similar manner if ,

Exercises

Find and for the following expressions: