$${\begin {array} {l} v = \frac {t^2 - {(t - \Delta t)}^2} {\Delta t} \\ = \frac {t^2 - t^2 + 2t\Delta t - \Delta t^2} {\Delta t} \\ = \frac {2t \Delta t - \Delta t^2} {\Delta t} \\ = 2t - \Delta t \end {array}}$$