Find $\mathbf{r}^{\prime} (t)$ and $\mathbf{r}^{\prime \prime} (t)$ for the vector function $\mathbf{r}(t)=e^{t}\mathbf{i}+\ln t\mathbf{j}-\cos t\mathbf{k}$.

Solution $$\begin{eqnarray*} \mathbf{r}^{\prime} (t) &=&\dfrac{d}{dt}e^{t}\mathbf{i}+\dfrac{d}{ dt}\ln t\mathbf{j}-\dfrac{d}{dt}\cos t\mathbf{k}=e^{t}\mathbf{i}+\frac{ 1}{t}\mathbf{j}+\sin t\mathbf{k} \notag \\[6pt] \mathbf{r}^{\prime \prime} (t) &=&\dfrac{d}{dt}e^{t}\mathbf{i}+ \dfrac{d}{dt}\frac{1}{t}\mathbf{j}+\dfrac{d}{dt}\sin t\mathbf{k}=e^{t} \mathbf{i}-\frac{1}{t^{2}}\mathbf{j}+\cos t\mathbf{k} \notag \end{eqnarray*}$$