Chapter VII · SUCCESSIVE DIFFERENTIATION.

LET us try the effect of repeating several times over the operation of differentiating a function (see p. 13). Begin with a concrete case.

Let .

There is a certain notation, with which we are already acquainted (see p. 14), 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 differential coefficient is , which is simpler to write than . This is called the "derived function" of .

Suppose we differentiate over again, we shall get the "second derived function" or second differential coefficient, which is denoted by ; and so on.

Now let us generalize.

Let .

But this is not the only way of indicating successive differentiations.

For, and this is more conveniently written as , or more usually .

Similarly, we may write as the result of thrice differentiating, .

Examples.

Now let us try .

In a similar manner if ,

Exercises IV. (See page 253 for Answers.)

Find and for the following expressions:

(1) .

(2) .

(3) .

(4) Find the 2nd and 3rd derived functions in the Exercises III. (p. 45), No. 1 to No. 7, and in the Examples given (p. 40), No. 1 to No. 7.