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.

 A: If $(a,b)$ is below the line $x=y$, then the function has value $y$ on a neighborhood of $(a,b)$, so the partial derivatives are 
$$\begin{align*}
\left.\frac{\partial f}{\partial x}\right|_{(x,y)=(a,b)}&=0\\
\left.\frac{\partial f}{\partial y}\right|_{(x,y)=(a,b)} &= 1.
\end{align*}$$
Symmetrically, if $(a,b)$ is "above" the line $x=y$, then the function has value $x$ on a neighborhood of $(a,b)$, so the partial derivatives are:
$$\begin{align*}
\left.\frac{\partial f}{\partial x}\right|_{(x,y)=(a,b)}&=1\\
\left.\frac{\partial f}{\partial y}\right|_{(x,y)=(a,b)} &= 0.
\end{align*}$$
If $(a,b)$ is on the line $x=y$, then the function has value $y$ as we approach along a constant $y$ direction from the right, and value $x$ if we approach along a constant $y$ direction on the left. So the partial with respect to $x$ is $1$ from the left and $0$ from the right, hence does not exist at $(a,b)$. Similarly for $y$.
So the function is differentiable away from the line $x=y$, with values as given above.
A: $$
f(x, y) = \min(x,y) = \begin{cases}
x & \text{if } x \le y \\
y & \text{if } x \gt y
\end{cases}
$$
The function isn't differentiable along $y = x$, but the partial derivatives are straightforward otherwise.
$$
\frac{\partial f(x, y)}{\partial x} = \begin{cases}
1 & \text{if } x \lt y \\
0 & \text{if } x \gt y
\end{cases}
$$
$$
\frac{\partial f(x, y)}{\partial y} = \begin{cases}
0 & \text{if } x \lt y \\
1 & \text{if } x \gt y
\end{cases}
$$
Here is a plot of the function to help you see the derivatives and why it's not differentiable along $y = x$:

A: 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.
