Question

if sqrt(y)=arctan(x), show that (1+x^2)dy/dx((1+x^2)dy/dx)=2

Answer 1

If

sqrt(y)=arctan(x),

that means
y = [arctan(x)]^2

Let u = arctan x and use the chain rule

y = u^2
dy/dx = d(u^2)/du * du/dx
= 2 arctan(x)/(1 + x^2)
or
(1 + x^2)*dy/dx = 2 arctan x = 2 sqrt y

Are you sure you wrote
(1+x^2)dy/dx((1+x^2)dy/dx)=2 correctly?
That has (1+x^2)dy/dx appearing twice in a row. It doesnt look right