Multivariable Chain rule Help

Question

a function $g(u)$ with continuous second derivative is given and f is defined by $f(x,y)=g(\frac{x}{y})$ for $y\neq 0$ how would you calculate $f_{yx}$.

I am very confused as to how to even approach this question since I am used to drawing dependency diagrams and working from there but I can't seem to make the connections for this question

Answer 1

First you find $f_y=g'(x/y)(-x/{y^2})$

This is a product and you want to differentiate with respect to $x$

$$f_{yx}=g''(x/y)(1/y)(-x/{y^2})+g'(x/y)(-1/{y^2})$$

Answer 2

Hint

$$f(x,y)=g\left(\frac{x}{y}\right)\implies f_x(x,y)=g'\left(\frac xy\right)\frac 1y.$$

$$f(x,y)=g\left(\frac{x}{y}\right)\implies f_y(x,y)=g'\left(\frac xy\right)\frac {-x}{y^2}.$$