1/2 Algebra. 



13. Examples. By the method already explained and illus- 

 trated find the derivative of the following expressions, supposing 

 in each case that a is the value of x from which the increase of x 

 is counted : 



1. 3.r-f2. 5. cx\ 



2. T,X^-\-2X. 6. C(x-^l)^. 

 J. (x+l)(x-\-2). ' 7. -'-. 



^. (x+c)\ 8. sj X. 



14. Extension of Meaning of D,J^'. 

 In Art. 10 we found that when 



j/=4^'-f5, D,y=8« 

 when y=^cx'^-\-e, T>^y=2ac 



and when j'=cx^-^ex-\-/, T)^y=2ac-\-e 



In each case D.^ r is of course a constant as it should be by the 

 definition in Art. 9, where T)_^y is defined to be a limit, and the 

 limit of a variable is by definition a constant. 



In each case here noticed J}^y is a constant whose value de- 

 pends upon the value a from which we begin to count the increase 

 of X, or, as we may say, D^y is a function of a, while in Art. 5 

 T)^y was a constant which does not depend upon a. 



In any case D,.^' is either a function of a or it is independent 

 of «, and when it is a function of a the a is the value from which 

 we begin to count the increase of x. 



Now, as we may begin to count the increase from any value of 

 jr, a is of course a7iy value of x, and so we may represent it by 

 X instead of a and relieve D^y from being a constant, or in- 

 other words, wherever T)^y was a function of a by the original 

 definition we regard it now and hereafter as the same function of 

 X that, by the original definition, it was of a. 



15. Let us now work out a case that was worked in Art. 11, 

 using X now where a was used before. 



Take y=^x^ + s- (0 



If we write x-\-Jx in place oi x and therefore y-\-M' in place of 

 y we have • 



j/+Jj=4r-r+Jx/ + 5. (2) 



