For x>1 , if (2 x)^ 2 y =4 e ^ 2 x-2 y , then (1+ 2 x)^2 d y d x is equal to

Step-by-step solution

(2 x)^ 2 y =4 e ^ 2 x-2 y Taking log on both sides, we get aligned & 2 y 2 x= (4 e ^ 2 x-2 y ) & 2 y 2 x= 4+ e ^ 2 x-2 y & 2 y 2 x= 4+2 x-2 y & 2 y 2 x+2 y= 4+2 x & 2(y 2 x+y)=2 2+2 x & y 2 x+y= 2+x & y(1+ 2 x)=x+ 2 & y= x+ 2 1+ 2 x aligned Differentiating w.r.t. x , we get gathered d y ~d x = (1+ 2 x)(1+0)-(x+ 2) ( 1 2 x ) 2 (1+ 2 x)^2 d y ~d x = (1+ 2 x)- 1 x (x+ 2) (1+ 2 x)^2 (1+ 2 x)^2 d y ~d x =1+ 2 x-1- 2 x (1+ 2 x)^2 d y ~d x = 2 x- 2 x = x 2 x- 2 x gathered