If $f(x, y) = x^2y + xy^3$ and $y^2+yx^2 = x^3$ , find expressions for the partial derivatives $∂f/∂x$ and $∂f/∂y$ and the total derivatives $df/ dx$ and $df/ dy$ in terms of $x$ and $y$. (In the latter you don’t need to simplify your answer.)
I've easily been able to calculate the partial derivatives of $f(x,y)$ like so:
$∂f/∂x = 2xy + y^3$, $∂f/∂y = x^2 + 3xy^2$
But what I am confused about is how to calculate the total derivatives? I would know how to calculate a total derivative if one variable is explicitly defined in terms of another, say t, but here I can't understand where to go? Can someone explain and help me?
Take the second equation. Using implicit differentiation, you can solve for $\frac{dy}{dx}$ and $\frac{dx}{dy}$. In the former, you treated $y=y(x)$, while in the latter, you say $x=x(y)$. Then by chain rule, $$ \frac{df}{dx}(x,y(x))=\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}\frac{dy}{dx}. $$ Similarly, $$ \frac{df}{dy}(x(y),y)=\frac{\partial f}{\partial y} + \frac{\partial f}{\partial x}\frac{dx}{dy}. $$