Yes. Multiplying any function by a constant changes its derivative by the same constant, so for any function f, if df = f then d(Cf) = Cf. So yes, you can multiply by C without losing the interesting properties of the derivative (whatever those might be), but you can do that to any function, and so the fact that you can do it to this function is true but not particularly interesting.
Comments
Note that C + Cx + ... would have also worked. So why the 1? Just simplicity?
Yes. Multiplying any function by a constant changes its derivative by the same constant, so for any function f, if df = f then d(Cf) = Cf. So yes, you can multiply by C without losing the interesting properties of the derivative (whatever those might be), but you can do that to any function, and so the fact that you can do it to this function is true but not particularly interesting.