Answer
$\dfrac{∂f}{∂x}|_{(0,-1,1)}=\text{Undefined}$ and $\dfrac{∂f}{∂y}|_{(0,-1,1)}=\text{Undefined}$
and $\dfrac{∂f}{∂z}|_{(0,-1,1)}=\text{Undefined}$
Work Step by Step
Given: $f(x,y,z)=\ln(x+y+z)$
We will find the partial derivatives as follows:
$\dfrac{∂f}{∂x}=\dfrac{1}{(x+y+z)}$ and $\dfrac{∂f}{∂y}=\dfrac{1}{(x+y+z)}$
and $\dfrac{∂f}{∂z}=\dfrac{1}{(x+y+z)}$
$\dfrac{∂f}{∂x}|_{(0,-1,1)}=\dfrac{1}{0-1+1}=\text{Undefined}$ and $\dfrac{∂f}{∂y}|_{(0,-1,1)}=\dfrac{1}{0-1+1}=\text{Undefined}$
and $\dfrac{∂f}{∂z}|_{(0,-1,1)}=\dfrac{1}{0-1+1}=\text{Undefined}$
