Let us try the effect of repeating several times over the operation of differentiating a function (see p. 14). Begin with a concrete case.

Let .

First differentiation, | . | |

Second differentiation, | . | |

Third differentiation, | . | |

Fourth differentiation, | . | |

Fifth differentiation, | . | |

Sixth differentiation, | . |

There is a certain notation, with which we are already acquainted (see p. 15), 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 .

First differentiation, | . |

Second differentiation, | . |

Third differentiation, | . |

Fourth differentiation, | . |

etc., etc. |

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

if the original function be ;

once differentiating gives ;

twice differentiating gives ;

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 255 for Answers.)
Find and for the following expressions:

- (1) .
- (2) .
- (3) .
- (4) Find the 2nd and 3rd derived functions in the Exercises III. (p. 46), No. 1 to No. 7, and in the Examples given (p. 41), No. 1 to No. 7.