Calculating partial derivatives of composition with max

Question

I want to calculate the partial derivatives of the function \begin{align*} \partial_i \left[ \frac{f(x)}{max(1,|f(x)|)} \right] ,i \in \{1,\dots,n\}. \end{align*} where $f: \Omega\subset \mathbb{R}^3 \to \mathbb{R}$ is smooth. Thank you.

Answer 1

In regions where $|f(x)| < 1$, your function is simply $f$, so the partial derivative is $\frac{\partial f}{\partial x_i}$.

In regions where $|f(x)| > 1$, your function is equal to the sign of $f(x)$, and its derivative is $0$.

On parts where $|f(x)|=1$, you have no guarantee that partial derivatives even exist. For example, if $f(x_1,x_2,x_3)=x_1$, then the partial derivative $\frac{\partial f}{\partial x_1}$ does not exist around the plane $x_1=1$