If y=[(x+1)(2 x+1)(3 x+1) .( n x+1)]^ n , then d y ~d x at x=0 is

Step-by-step solution

aligned & y=[(x+1)(2 x+1)(3 x+1) ( n x+1)]^ n & y= nlog [(x+1)(2 x+1)(3 x+1) & y= n [ (x+1)+ (2 x+1) & + (3 x+1)+ + ( n x+1)] aligned Differentiating both sides w.r.t. x , we get aligned & 1 y d y ~d x = n ( 1 x+1 + 2 2 x+1 + 3 3 x+1 + + n n x+1 ) & 1 1 ( d y ~d x )_ x=0 = n (1+2+3+ + n ) aligned [ At x=0, y=1] ( d y ~d x )_ x=0 = n [ n ( n +1) 2 ]= n ^2( n +1) 2