Question

Compute \(\mathbf{r}^{\prime \prime}(t)\) and \(\mathbf{r}^{\prime \prime \prime}(t)\) for the following functions. $$\mathbf{r}(t)=\tan t \mathbf{i}+\left(t+\frac{1}{t}\right) \mathbf{j}-\ln (t+1) \mathbf{k}$$

Short answer

Question: Find the second and third order derivatives of the vector function \(\mathbf{r}(t) = (\tan t) \mathbf{i} + (t + \frac{1}{t})\mathbf{j} + (-\ln(t + 1))\mathbf{k}\). Answer: The second and third order derivatives of the given vector function are: \(\mathbf{r}^{\prime \prime}(t) = (2\sec^2 t \tan t) \mathbf{i} + (2t^{-3})\mathbf{j} + \frac{1}{(t+1)^2}\mathbf{k}\) and \(\mathbf{r}^{\prime \prime \prime}(t) = (4\sec^2 t(\tan^2t+1)) \mathbf{i} + (-6t^{-4})\mathbf{j} + \frac{-2}{(t+1)^3}\mathbf{k}\).

Step-by-step solution

Derive \(\mathbf{r}^{\prime}(t)\) for each component

To find the first derivative, compute the derivative with respect to \(t\) for each component of the vector separately: $$\mathbf{r}^{\prime}(t) = \frac{d}{dt}(\tan t \mathbf{i}) + \frac{d}{dt}(t+\frac{1}{t})\mathbf{j} + \frac{d}{dt}(-\ln(t+1))\mathbf{k}$$

Compute the derivatives for each component in \(\mathbf{r}^{\prime}(t)\)

Calculate the derivatives of each component: $$\mathbf{r}^{\prime}(t) = (\sec^2 t) \mathbf{i} + (1-\frac{1}{t^2})\mathbf{j} + \frac{-1}{t+1}\mathbf{k}$$

Derive \(\mathbf{r}^{\prime \prime}(t)\) for each component

Compute the second derivative with respect to \(t\) for each component of the vector separately: $$\mathbf{r}^{\prime \prime}(t) = \frac{d^2}{dt^2}(\sec^2 t \mathbf{i}) + \frac{d^2}{dt^2}(t-\frac{1}{t})\mathbf{j} + \frac{d^2}{dt^2}(-\frac{1}{t+1})\mathbf{k}$$

Compute the derivatives for each component in \(\mathbf{r}^{\prime \prime}(t)\)

Calculate the derivatives of each component: $$\mathbf{r}^{\prime \prime}(t) = (2\sec^2 t \tan t) \mathbf{i} + (2t^{-3})\mathbf{j} + \frac{1}{(t+1)^2}\mathbf{k}$$

Derive \(\mathbf{r}^{\prime \prime \prime }(t)\) for each component

Compute the third derivative with respect to \(t\) for each component of the vector separately: $$\mathbf{r}^{\prime \prime \prime}(t) = \frac{d^3}{dt^3}(2\sec^2 t \tan t \mathbf{i}) + \frac{d^3}{dt^3}(2t^{-3}\mathbf{j}) + \frac{d^3}{dt^3}(\frac{1}{(t+1)^2}\mathbf{k})$$

Compute the derivatives for each component in \(\mathbf{r}^{\prime \prime \prime}(t)\)

Calculate the derivatives of each component: $$\mathbf{r}^{\prime \prime \prime}(t) = (4\sec^2 t(\tan^2t+1)) \mathbf{i} + (-6t^{-4})\mathbf{j} + \frac{-2}{(t+1)^3}\mathbf{k}$$ So, the second and third order derivatives of the given vector function are: $$\mathbf{r}^{\prime \prime}(t) = (2\sec^2 t \tan t) \mathbf{i} + (2t^{-3})\mathbf{j} + \frac{1}{(t+1)^2}\mathbf{k}$$ $$\mathbf{r}^{\prime \prime \prime}(t) = (4\sec^2 t(\tan^2t+1)) \mathbf{i} + (-6t^{-4})\mathbf{j} + \frac{-2}{(t+1)^3}\mathbf{k}$$