How does an absolute value operator interact with partial derivatives?

Question

I wish to take the partial derivative with respect to $y$ of the following expression:

$$g(x,y)=\log{\left|\frac{\partial}{\partial x}f(x,y)\right|}$$

where $f(x,y):\mathbb{R}^2\rightarrow\mathbb{R}$ is some function which depends on $x$ and $y$, and $\frac{\partial}{\partial x}$ denotes the partial derivative with respect to $x$. How does the absolute value operator $|\cdot|$ interact with the partial derivative $\frac{\partial}{\partial y}$ if I want to evaluate $\frac{\partial}{\partial y}g(x,y)$?

I would greatly appreciate any suggestions or references on how to address such situations, also in more general cases.

Answer 1

Assuming that $\frac{\partial }{\partial x}f(x,y)\ne 0$ everywhere, by the chain rule, you need the existence of partial derivative with respect to $y$ for the map $$ y\mapsto |\frac{\partial }{\partial x}f(x,y)| $$

Answer 2

Write $$g(x,y)=\log|f_x|.$$ Then you have that $$g_y=\frac{1}{|f_x|}\frac{f_x}{|f_x|}f_{xy}=\frac{f_{xy}}{f_x}.$$