Is $c \frac{\partial}{\partial x}=\frac{\partial c}{\partial x}=0$? Is $c \frac{\partial}{\partial x}=\frac{\partial c}{\partial x}=0$?

Or is one supposed to treat $c \frac{\partial}{\partial x}$ as "partially applied" operator (which expects a function).
I don't see how "linearity" of the derivative, that $\frac{\partial}{\partial x}(af(x)+ bg(x))= a\frac{\partial f}{\partial x}+ b\frac{\partial g}{\partial x}$, has anything to do with this. It is just a matter of notation- understanding what $c\frac{\partial}{\partial x}$ means as differential operator.
If c is a number , yeah thats right 
We know that Differentiate of every number relative to x is equal to 0 so the second side is equal to 0 .
(differentiate of c and 1 relative to x is 0) so in the left side we have : [c(0)=0] 
So yeah the equation is true.

Attention: my original language is not english and im not proffesional on it . so excuse me if i have mistake in my writing or in my words)