## 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.