Equal derivatives

Question

Does there exist a nonconstant function such that $\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}$? How about $\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}$?

Edit: I should say this is not for homework. I just thought of these two problems. I was wondering if they are true.

Answer 1

Yes: $f=x+y+z+C$ and $f=x+y+2z+C$ .

Answer 2

For your first question:

$f(x,y,z)=e^{x+y+z}$ -- (in fact, this one satisfies $f=\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z})$

and your second question: $f(x,y,z)=e^{x+y+2z}$