what is the implicit differentiation of (xy)^2 +3y^2=5?

Question

Please help, i'm stuck.

Answer 1

x^2y^2+3y^2=5 when differentiated becomes 2xy^2+2x^2ydy/dx+6ydy/dx=0. 2 can be removed as a common factor and dy/dx=-xy^2/(yx^2+3y)=-xy/(x^2+3).

Answer 2

Given (xy)^2 +3y^2=5

Implicit differentiation of
x^2y^2+3y^2=5
x^2 2yy'+y^22x+6yy'=0
x^2 2yy'+6yy'=-2xy^2
2yy'[x^2+3y]=-2xy^2
y'=-[xy/(x^2+3)]

Implicit Differentiation

Answer 3

Let f(x,y)=x²y²+3y²-5=0 ···Eq.1

f(x,y)=0, so ∂f/∂x + (∂f/∂y)dy/dx = 0

Thus, dy/dx=-(∂f/∂x)/(∂f/∂y) ··· Eq.2

From Eq.1, ∂f/∂x=2xy², ∂f/∂y=2x²y+6y

From Eq.2, dy/dx=-2xy²/(2x²y+6y)=-2xy²/[2y(x² +3)]=-xy/(x²+3)

The answer is: dy/dx=-xy/(x²+3)