i8o Algebra. 



It is to be noticed that this formula appHes to the derivative 

 with respect to x of a power of x. Of course any other letter be- 

 sides X could be used to denote a variable. 



Thus, wheny=a"", T>^j'=??a2"~\ 



But we must be careful not to use this formula to find the de- 

 rivative with respect to some quantity, of a power of some ofker 

 quantit3% Or, in oth,er words, in order to be able to use this form- 

 ula the quantity which is raised to a power must be same as that 

 with respect to which the derivative is taken. 



28. I^o FIND THE Derivative with Respect to x of any 

 Negative Integral Power of x. 



Let j^=x-"=^. (i) 



Do^="^"^-'~«' ^^ by Art, 23. (2) 



Simplifying, remembering that the derivative of a constant is 

 zero, we get 



It may be objected to this method that we have used the form- 

 ula for the derivative of a fraction whose numerator is i when 

 that formula supposed that numerator and denominator were 

 each functions of x. 



We may then take y= "„^_^ 



and 7102V use the formula of Art. 23 and we get as before 



D^.y= — nx~''~\ 



It easily follows that if 



y=ax~'\ 

 then D^y=—riax~"~\ 



Here, as in Art. 27, in order to use the formula the quantity 

 raised to a power must be the same as the one with respect to 

 which the derivative is taken. 



We may express this formula in words thus — 



The derivative ivith respect to x of ax~" is found by yniiltiplying 

 the exponent by the coefficient and reduci7ig the exp07ient i. 



