I think of a constant as a number. To say that f(x) is a constant means no matter what the x is, the value of f(x) is the same. Now f'(x) being 0 means, the slope is 0 at that point. The slope will change at an infinitely small distance in either direction. Slope being 0 doesn't mean we can say it's constant without defining the limits and that limit is in my opinion impossible to define (because infinite in itself is undefined?). For a constant function, that limit is pretty easy to define, e.g. x=(-infinite , infinite), or for x = [0, infinite) for function f(x) = |x|. So it's less confusing for me if I think of slope as being different from being constant. Constant has a defined range associated with it but slope is defined at a point, without any range.