Derivatives. 167 



responding increase in the value of v it is well to have a conven- 

 ient notation by which these amounts of increase are denoted. So 

 in future we will use Jx to denote the increase in the value of x 

 and J I' to denote the correspondrng increase in the value of r. 



In this notation the fraction at the end of Art. 5 would be 

 written 



The student is cautioned not to think of J.v as being J times 

 X, for the symbol J as here used does not stand for a quantity at 

 all, but simply for the words increase of. 



7, Let us now consider the equation 



J'=-v'+i. 

 In this equation give x the successive integer values from —3 

 to 7 and compute the corresponding values of y. We may ar- 

 range the results as in Art. 5, 



y_ \ iQ 5 2 I 2 5 10 17 26 37 5 

 v I —3 —2 —I o I 2 3 4 5 6 7 

 If x=i the corresponding value oi y is 2, and if ^==7 the cor- 

 responding value of J is 50, and if x be supposed to increase 

 from I to 7, at the same time y will increase from 2 to 50, or 

 starting at x=\, if x increase by 6 then y will increase by 48, or 

 when Ja-=6, Jj/=48. 



Still starting at .r= i , let us give to J-r various values and de- 

 termine the corresponding values of Jj'. 

 The results may be arranged in the form 



jj_| 48 35 24 15 8 3 

 Jjt- I 6 5 4 3 2 1- 



Ay 

 Here we have a case where the ratio j- is not always the same 



as it was in Art. 5, but at one time it is Y"* o^ ^' ^t another time 

 it is -V-, or 7, etc. 



As can be seen by the above scheme, the fraction -y takes suc- 

 cessively the values 8, 7, 6, 5, 4, 3 as J.v takes '^the successive 

 values 6, 5, 4, 3, 2, i. 



We now give to x values intermediate between i and 2 and 



