90 PRACTICAL MATHEMATICS 



(c) To differentiate a product. 



If y = uv where u and v are functions of x, 

 then y + By = (u + Su) (v + Bv) 



= uv + u Bv + v Bu + Bu Bv 

 and %y = u Bv + v Bu + Bu Bv 



By Bv Sw ~ Bv 

 /- = u Y - + v^- + Su T - 

 Bx ox ox ox 



Making $x infinitely small, 



dy _ dv du 



dx dx dx 



Since Sw becomes infinitely small along with &#, the term 



Sw -T- becomes negligibly small when Bx is made infinitely small. 



CiCC 



To differentiate x n n x , 



y = x n n x = u v 



fj>it 



Then u = x n and -^- = nx n ~ 1 



dx 



v = n x and -=- = n x log g n 



du dv du 



but -~ = u -j- + v -j- 



dx dx dx 



= x n n x log e n + n x nx 11 - 1 

 = n x x n ~ 1 (x log e n + n) 



If^ = M-y-+u-3- when y = uv, dividing throughout by y we get 



1 dy 1 du 1 dv 



y dx u dx v dx 



Sometimes this second form is more easily worked with than 

 the first form. Also each term on the right-hand side is built 

 up from one term of the product only. For example, the first 

 term only contains u, the second term only contains v, and there- 

 fore we can extend the result to suit the case when the product 

 contains any number of factors. 



Thus if y = u v w, 



I dy I du I dv 1 dw 



then ~:r = --r + -j- + - T~ 



y dx u dx v dx w dx 



(d) To differentiate a fraction. 

 Let y=- 



then 



