If y=(1+x) (1+x^2 ) (1+x^4 ) (1+x^ 2 n ) , then ( d y d x )_ x=0 is equal to

Step-by-step solution

Given aligned y & = (1+x) (1+x^2 ) (1+x^4 ) (1+x^ 2 n ) (1-x) (1-x) & = (1-x^2 ) (1+x^2 ) ( (1+x^4 ) (1+x^ 2 n ) . 1-x y & = [1- (x^ 2 n )^2 ] 1-x = 1-x^ 4 n 1-x aligned aligned d y d x & = (1-x) (-4 n x^ 4 n-1 )- (1-x^ 4 n )(-1) (1-x)^2 ( d y d x )_ x=0 & = 1(-0)-(1-0)(-1) 1^2 & = 1 1 =1 aligned