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

Step-by-step solution

y=[(x+1)(2 x+1)(3 x+1) ( n x+1)]^ 3 2 Taking 'log' on both sides, we get array r y= 3 2 [ (x+1)+ (2 x+1)+ (3 x+1) + + (n x+1)] array Differentiating w.r.t. x , we get aligned & 1 y d y ~d x = 3 2 [ 1 x+1 + 2 2 x+1 + 3 3 x+1 + + n n x+1 ] & d y ~d x = 3 y 2 [ 1 x+1 + 2 2 x+1 + 3 3 x+1 + + n n x+1 ] & Now at x=0, y=[ (1)(1)(1) (1) _ n times ]^ 3 2 =1 aligned aligned . d y ~d x |_ x=0 & = 3(1) 2 [ 1 0+1 + 2 0+1 + 3 0+1 + + n 0+1 ] & = 3 2 (1+2+3+ + n ) & = 3 2 n ( n +1) 2 & = 3 n ( n +1) 4 aligned