Find the partial derivatives of second order of $f(x,y)=\varphi(xy,\frac{x}{y})$

Question

Ok guys, I'm given this smooth function $\varphi(u,v)$ defined in $R^2$. So that $f(x,y)=\varphi(xy,\frac{x}{y})$. I have to find all partial derivatives of second order of $f$ using the partial derivatives of $\varphi$. I know how to find the partial derivative of "normal" functions like $\frac{xy}{x+y}$ or something like that, but this kind of problem I have no idea how to do. Any ideas and solutions are welcomed $\ddot \smile $

Answer 1

Use the chain rule: $$ \frac{\partial}{\partial x} \phi(u(x,y),v(x,y)) = \frac{\partial \phi}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial \phi}{\partial v} \frac{\partial v}{\partial x}, $$

$$ \frac{\partial}{\partial y} \phi(u(x,y),v(x,y)) = \frac{\partial \phi}{\partial u} \frac{\partial u}{\partial y} + \frac{\partial \phi}{\partial v} \frac{\partial v}{\partial y}. $$