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\begin{gathered} \frac{\text { change in position }}{\text { change in time }}=\frac{\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}}{\Delta t}=\sqrt{\frac{(\Delta x)^{2}+(\Delta y)^{2}}{(\Delta t)^{2}}} \\ \qquad \qquad \qquad \qquad =\sqrt{\left(\frac{\Delta x}{\Delta t}\right)^{2}+\left(\frac{\Delta y}{\Delta t}\right)^{2}} \\ \quad \qquad \qquad \qquad \qquad \qquad \qquad \rightarrow \sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}} \qquad \text { as } \Delta t \rightarrow 0 \end{gathered}