Given: #sqrt(5)x + y = 3 + x^2y^2#
Use the product rule #(uv)' = uv' + vu'#
Let #u = x^2, u' = 2x " and "# Let # v = y^2, v' = 2y y'#
Implicit differentiation:
#sqrt(5) + y' = 0 + x^2(2yy') + y^2(2x)#
#sqrt(5) + y' = 2x^2y y' +2xy^2#
Get the #y'#s on the same side of the equation and everything else on the other side of the equation:
#y' - 2x^2y y' = 2xy^2 - sqrt(5)#
Factor the #y'#:
#y' (1 - 2x^2y) = 2xy^2 - sqrt(5)#
Isolate #y'# by dividing:
#y ' = (2xy^2 - sqrt(5))/(1 - 2x^2y)#
or #" "y ' = -(2xy^2 - sqrt(5))/(2x^2y-1)#
