Some Applications of the Chain Rule

If we know the position of an object at every time, then we can determine its speed. The formula for speed comes from the distance formula and looks a lot like it, but with derivatives.

Proof: The speed of an object is the limit, as $\Delta \mathrm{t} \rightarrow 0$, of $\frac{\text { change in position }}{\text { change in time }}$. (Fig. 6 )

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

Exercise 10: Find the speed of the object whose location at time $\mathrm{t}$ is $(\mathrm{x}, \mathrm{y})=\left(\sin (\mathrm{t}), \mathrm{t}^{2}\right)$ when $\mathrm{t}=0$ and $\mathrm{t}=1$.

Solution: $\mathrm{dx} / \mathrm{dt}=\cos (\mathrm{t})$ and $\mathrm{dy} / \mathrm{dt}=2 \mathrm{t}$ so speed $=\sqrt{(\cos (\mathrm{t}))^{2}+(2 \mathrm{t})^{2}}=\sqrt{\cos ^{2}(\mathrm{t})+4 \mathrm{t}^{2}}$.

Practice 7: Show that an object whose location at time $t$ is $(x, y)=(3 \sin (t), 3 \cos (t))$ has a constant speed. (This object is moving on a circular path).

Practice 8: Is the object whose location at time $\mathrm{t}$ is $(\mathrm{x}, \mathrm{y})=(3 \cos (\mathrm{t}), 2 \sin (\mathrm{t}))$ travelling faster at the top of the ellipse ( at $\mathrm{t}=\pi / 2$ ) or at the right edge of the ellipse (at $\mathrm{t}=0$)?