I am having a hard time understanding the following:
I need to have $\frac{dR}{dS}$, if I am given the following equations in terms of $x$ and $y$.
$R(x,y)= x^3 + xy$ and $S(x,y)=6x^2y$.
I was thinking of using the following: $\frac{dR}{dS} = \frac{\partial R}{\partial x}\frac{dx}{dS} + \frac{\partial R}{\partial y}\frac{dy}{dS}$. Is this correct?
For example if I took the derivative of $S$ with respect to $x$, in order to get the reciprocal of $\frac{dx}{dS}$, wouldn't I end up with the term $\frac{dy}{dx}$ because of the chain rule? I think I'm mostly confused on obtaining the terms $\frac{dx}{dS}$ and $\frac{dy}{dS}$ if I only have one equation involving $S$.