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I'm just wondering if you have a distance function: $$ s(t) = 0.1t^2 - 5t $$ where $s(t)$ is distance and $t$ is time in seconds, does differentiating it give you a function for velocity?

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Distance is a scalar quantity. Differentiating distance with respect to time gives speed. Speed is also a scalar quantity. While velocity is a vector quantity, and velocity is the differentiation of displacement with respect to time. Displacement is a vector quantity.

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Yes, it does. The average velocity over a period $\Delta t$ is given by $$ v = \frac{\Delta s}{\Delta t} $$ The (instantaneous) velocity is the average velocity upon an infinitesimal interval of time $$ v = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t} = \frac{ds}{dt} $$ The latter equality follows immediately from the definition of a derivative.

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