104 PRACTICAL MATHEMATICS 



(a) To expand (x+ y) n 



Then z = (x + y} n ; and when y = 0, z = x n , and this gives A as 

 a function of x. 



If A = x n 



d r A. 



-7-7= n(n- l)(n- 2) . . . (w-r+l) n - f 



dA 



when r = 1 -= = wa; 71 - 1 

 02; 



r = 2 -j-- = n(n l)x n ~ 2 , and so on. 



But z = (a; + y) n 



dA y 2 d 2 \ y 

 ^ + + 



-- 



I i 



(&) To expand log e ( + y) 



Then s = log e (# + y) ; and when y = 0, s = log^, and this gives 

 A as a function of x. 



If A = 



x 3 



rfn A I W 1 



u A = / _ i\n ' 



dx n v ; x n 

 But 2 = loge(a; + y) 

 dA 



