ELISHA MITCHELL SCIENTIFIC SOCIETY. 1 5 



respect to x as the independent variable. We have hitherto 

 supposed X to have a constant vahie, but the above method 

 of finding the derivative is the same whatever value of .r, 

 o-ivino- real values to r in o, we start from; hence the 

 method is perfectly general. 

 As an illustration let, 



y=f{x)= X^ + b. 



If X is changed to (.r ^ h) = {x — A.r), y will be changed 

 to J 4- A_>'. 



. • . 7 + a;' = /(.r -^ /i) = ix -f /if -r /i. 

 Expanding the right member of the last equation and sub- 

 tracting the preceding equation from it, we have, 

 A>' = ^x-/i — 7,x/r -j- k^. 

 Dividing by /i = ax, we have, 



AJ' 



— = 3't" + iS'^' + ^-'^) ^-^' (4)- 



A.l' 



The limit to which the right member approaches indefi- 

 nitely is 3-1", since as a.i' diminishes indefinitely, so does 

 the term (3^ -f ax) ax in which A.r is a factor. Therefore 

 the derivative of /(.f) = x^ — ^ with respect to x is, 



Aj' /(x -f A.r) — /(:t-) 



lim. — = lim. = ^x^. 



A.r A-i" 



This limit (3.1'-) is true, no matter what value of x we 

 start from, and its numerical value depends upon the value 

 of X. It is seen to be perfectly definite and finite and to 

 varv from zero to plus infinity according as x changes from 

 zero to infinity. For a given value of x as 2, the limit 

 has only one value = 12. Similarly for any other value. 

 It is only in the case of the simpler functions that/'(.f — /i) 

 can be developed readily, so that the derivatives can be 

 easily found, but after rti/es for finding the derived func- 

 tions of products, powers, etc., have been deduced (as given 

 in elementarv treatises on the calculus) the work of finding 



