Derivative of the $f(x,y)=\min(x,y)$ I just encountered this function $f(x,y)=\min(x,y)$. I wonder what  the partial derivatives of it look like.
$$
f(x, y) = \min(x,y) = \begin{cases}
x &amp; \text{if } x \le y \\
y &amp; \text{if } x \gt y
\end{cases}
$$

The function isn't differentiable along $y = x$, but the partial derivatives are straightforward otherwise.

$$
\frac{\partial f(x, y)}{\partial x} = \begin{cases}
1 &amp; \text{if } x \lt y \\
0 &amp; \text{if } x \gt y
\end{cases}
$$

$$
\frac{\partial f(x, y)}{\partial y} = \begin{cases}
0 &amp; \text{if } x \lt y \\
1 &amp; \text{if } x \gt y
\end{cases}
$$

Here is a plot of the function to help you see the derivatives and why it's not differentiable along $y = x$:
It refers to the partial derivatives of functions of two variables or more/simply multiple variables,with their notation $f\{x,y,z,.....\}$, depending on how many variables are given.this method can be done by differentiating with respect to individual variable (with respect to $x$,you fix $y$ and $z$ and treat them as constants, with respect to $y$, you fix $x$ and $z$ and treat them as constants and the same goes for $z$ $zd_n$ other given variables. We can also use the formula.