20 DIFFERENTIAL COEFFICIENT OF A QUOTIENT. 



31. Differential coefficient of a quotient. 



Let <j>(x) and ty(x) denote two functions of x, and let 



Suppose x changed into x + h, and let u + AM denote the 

 new value of the quotient. Then 



therefore AM = <^ + *>*/*> T*<f 



i/r (x + A) i/r (x) 



(x) - ty (x + h) ~^(x}} <^> (a?) . 

 (x) 



. i p LA /i 



therefore -r = 

 Aa; 



Let h diminish without limit, then 



du _ <f> (x) T/r (x) -v|r' (a;) <j> (x) 



dx~ tyW 



Hence we have this rule : To find the differential coefficient 

 of a quotient ; multiply the denominator by the differential 

 coefficient of the numerator and the numerator by the differential 

 coefficient of the denominator; subtract the second product from 

 the first and divide the result by the square of the denominator. 



32. The result of Art. 31 may also be obtained thus : 



<f>(x) 

 Since u = ---. , 



therefore < (x) = u 



therefore, by Art. 29, 



therefore fW 



du 

 therefore -- = 



