How can we prove that the derivative is the best approximation of rate of change?

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I am a Physics Student so learning about derivatives is most essential, but how can you use this "derivative" to calculate Speeds and other such things accurately just by applying limits.

Note: I already know about epsilon-delta definition and the standard definition of a derivative, the graphic interpretation too but this doesn't satisfies me enough that how can I use the construct such as "derivative" for studying motion of objects and other such things.