Multivariable Chain Rule - How to solve this?

Question

$g(x,y) = f(x^2 - y^2, 2xy)$

How do I find the the partial derivative of $g$ with respect to $x$ (in terms of $f$) in this case?

Thanks for the help! I honestly can't figure out how to do this :(

Answer 1

Let $u=x^2-y^2$, $v=2xy$. Then, $g(x,y)=f(u,v)$, and \begin{align} g_x &= \frac{\partial f}{\partial x}\\ &=\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial x}\\ &=2xf_u+2yf_v\\ \end{align}