CHANGE OF THE INDEPENDENT VARIABLE. 177 



_ 

 ?f \dy \dy \df 



- 



dy 



(-Y 



\d) 



d 3 x dx 



~dfdy 



This process is called changing the independent variable 

 from x to y. 



199. With respect to the use of the preceding Articles 

 we must observe that, as is the case with some other parts 

 of the Differential Calculus, the student is here acquiring 

 materials which will be available in some of his following 

 subjects. Expressions which present themselves can some- 

 times be much simplified by transforming them in the manner 

 above indicated ; of this examples will be seen at the end 

 of this Chapter. 



200. The following is an important special case. 



d n y 

 Change the independent variable in x n -^ from a; to t, 



where x = e*. 



d f d n y\ d f n d n y\ dx 



We have -y (x ~ n ) = T- ( x -r^. -si 



dt \ dx J dx \ dx J dt 





dx n 



This result may for the sake of abbreviation be thus ex- 

 pressed, 



(d \ dry 

 (dr n ) X dx = ' 



T. D. C. N 



