how do you estimate the instantaneous rate of change

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how do you estimate the instantaneous rate of change

Mathematics
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first i made of tables of points that are getting closer and closer the point but im not sure what to do next
if you are evaluating "instantaneous" rate of change, you'll evaluate your function as t (time) goes to zero, i think u need to use limits \[\lim_{t \rightarrow 0}(f(t)/t)\]
so my equation is f(x)=(100x^2)/(t^(3)+5t^(2)-100x+380) and in want to know the instantaneous velocity at f(10) i would take the limit of that equation as x->10?

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\[f'(10) \text { \to find instaneous rate of change at x=10}\]
you have x and t going on there that is weird
oh ya the t's are supposed to be x's
Instantaneous rate for change is\[f'(x)=\lim_{\Delta x \rightarrow 0}{f(x+\Delta x)-f(x)\over\Delta x}\]so the estimation is the same formula with a finite sized Delta x; i.e. no limit\[f'(x)=\lim_{\Delta x \rightarrow 0}{f(x+\Delta x)-f(x)\over\Delta x}\approx{f(x+\Delta x)-f(x)\over\Delta x}\]for reasonably small Delta x. How small Delta x has to be for a good estimate depends of how curved the function is around the point in question.

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