190 Algebra. 



Taylor's formula. 



13. Taylor's formula is a very general one that enables us to 

 obtain the development of a function of a binomial x-\-h arranged 

 according to positive increasing powers of //. Whether the func- 

 tion be integral or fractional, rational or irrational, it matters not ; 

 indeed any func':ion of a binomial x+h which is capable of being 

 expressed in the form of a series arranged according to positive 

 increasing powers of // can be thus expressed by means of Ta3'lor's 

 formula. 



Sometimes the series will be finite and sometimes (indeed 

 usually) infinite, but in case the series is infinite it must be re- 

 membered that the series and the function cannot be considered 

 equivalent unless the series is convergent. 



Before we can take up Taylor's formula it is necessary to ex- 

 plain what is meant by successive derivatives and to give a theorem 

 not given in the chapter on derivatives. These we now take up. 



14. Succkssive: DerivativEvS. If we represent a function of x 

 hyf(x) w^e may find the derivative with respect to -v of this func- 

 tion of A", and, as the result is usually another function of x, .we 

 may represent it by f'(x). Again, we may find the derivative 

 with respect to x oif'(x) and may represent this by f.'(x). 



Thus we see that f"(x) is the derivative wath respect to x of 

 the derivative with respect to x of /(^.vj. This is- called the second 

 derivative ivith respect to x of f(x), and is represented by the nota- 

 tion Dlfix). 



We may find the derivative with respect to .v oi/"(x) and rep- 

 resent the result by f"'(x). Thus we say that f"'(x) is the 

 derivative with respect to x of the second derivative with respect 

 to X oi f(x). This is called the third derivative with respect to x 

 oif(x), and is represented by the notation V>%f(x), and so qu. 



For example, if we take ax" as the function of .i: we start with, 

 then 



T>^ax"=nax"~'^ 



Dlax"=T>,nax"-'=7i(n—i)ax"-' 

 T>f.ax"=D,,-n(n— I )ax"~''=n( n— I J(n — 2)ax"~^ 

 etc. 



