Calculus confusion

Calculus: An Intuitive and Physical Approach by Morris Kline 1977 had something that help me clarify notational issues.  On pg 26 he writes:

(22) \lim \limits_{\Delta t \to 0}{ \Delta s \over \Delta t} = 178

Even this notation in (22) is somewhat lengthy.  Hence mathematicians replace it by a still briefer one.  Newton used \dot s .  Leibnex used ds \over dt .  Still another notation s' was introduced by Lagrange and if Euler’s notation f(t) is used then the limit (22) is denoted f'(t) .  Thus

(22) \lim \limits_{\Delta t \to 0 }{ \Delta s \over \Delta t} = \dot s = {ds \over dt} = s' = f'(t)

The four notations, are identical in meaning by all fall short of perfection.  Newton’s symbolism is concise but poor because the dot above the s is often overlooked;  moreover it fails to show what the independent variables is.  Leibniz’s notation suggests that the limit is a quotient whereas the limit of average speeds definitely is not a quotient … The notation f'(t) is informative but clumsier than \dot s or s' .  … However we shall only use the dot notation when the independent variable is time.

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