Ok, so we define derivatives as functions that gives us almost instantaneous rate of change. Using limits we calculate the derivatives as single functions. Like in the case of finding average velocity between t1 and t2, Avg v = [s(t2)-s(t1)]÷[t2-t1] but here we have two independent variables t1 and t2, we get rid of it during finding derivatives by using h (h approaches 0). So we can also write the average change as a function by defining a constant 'a' such that average velocity AROUND TIME 't' can be calculated as [s(t+a)-s(t-a)]÷2a. I was wondering if this already is a thing in mathematics or it is just a Bunch of nonsense.
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2$\begingroup$ Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. $\endgroup$José Carlos Santos– José Carlos Santos2024-01-03 09:18:45 +00:00Commented Jan 3, 2024 at 9:18
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Yes, that’s a thing. The limit of that quotient is called the symmetric derivative; if it exists, the function is called symmetrically differentiable at $t$. Differentiability implies symmetric differentiability but not vice versa. For instance, the absolute value function $s(t)=|t|$ is symmetrically differentiable at $t=0$, with symmetric derivative $0$, but it’s not differentiable at that point.
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1$\begingroup$ Thanks man, that genuinely boosts my creativity towards mathematics. $\endgroup$Vikas Asdev– Vikas Asdev2024-01-15 17:32:31 +00:00Commented Jan 15, 2024 at 17:32