Find the derivative of each vector function.

(a) $\mathbf{r}(t)=2\sin t\mathbf{i}+3\cos t\mathbf{j}$

(b) $\mathbf{r}(t)=e^{t}\mathbf{i}+(1+t)\mathbf{j}+\sin t\mathbf{k}$

Solution

(a) We find the derivative of each component. Since $\dfrac{d}{dt}\left( 2\sin t\right) =2\cos t$ and $\dfrac{d}{dt}\left( 3\cos t\right) =-3\sin t,$ $$\begin{equation*} \dfrac{d\mathbf{r}}{dt}=\mathbf{r}^\prime (t)=2\cos t\mathbf{i}-3\sin t \mathbf{j} \end{equation*}$$

(b) $\dfrac{d\mathbf{r}}{dt}=\mathbf{r}^\prime (t)=\dfrac{d}{dt} e^{t}\mathbf{i}+\dfrac{d}{dt}(1+t)\mathbf{j}+\dfrac{d}{dt}\sin t \mathbf{k}=e^{t}\mathbf{i}+\mathbf{j}+\cos t\mathbf{k}$