Let f(x,y)=y³+10xy+4x³-12x ··· Eq.1
Since f(x,y)=0, the total derivative, with respect to x, of both sides of f(x,y)=0 can be written as follows:
∂f/∂x+(∂f/∂y)(dy/dx)=0 therefore, dy/dx=-(∂f/∂x)/(∂f/∂y) ··· Eq.2
From Eq.1, ∂f/∂x=10y+12x²-12, and ∂f/∂y=3y²+10x Thus, from Eq.2, we have:
dy/dx=-(10y+12x²-12)/(3y²+10x)=-2(5y+6x²-6)/(3y²+10x)
The answer is: dy/dx=-2(5y+6x²-6)/(3y²+10x)