Find dy/dx by implicit differentiation? √5x + y = 3 + x^2y^2

1 Answer
Apr 28, 2017

#y' = (2xy^2-sqrt(5))/(1-2x^2y)#

or #" "y ' = -(2xy^2 - sqrt(5))/(2x^2y-1)#

Explanation:

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)#