12.2q3 ### Problem: Finding the Derivative of a Vector-Valued FunctionGiven the vector-valued function:\[ \mathbf{r}(t) = 8 \sqrt{t} \mathbf{i} + t^2 \sqrt{t} \mathbf{j} + \ln(t^2) \mathbf{k} \]We are tasked with finding the derivative \(\mathbf{r}'(t)\).### Steps to Find the Derivative \(\mathbf{r}'(t)\):1. **Differentiate each component of the vector function with respect to \(t\):** - \[\mathbf{r}(t) = 8\sqrt{t} \mathbf{i}\] - \[\mathbf{r}(t) = t^2 \sqrt{t} \mathbf{j}\] - \[\mathbf{r}(t) = \ln(t^2) \mathbf{k}\] 2. **Apply the differentiation rules for each component:** - For **\( \mathbf{i} \)-component**: - The derivative of \( 8\sqrt{t} \): - \( \frac{d}{dt}\left(8\sqrt{t}\right) = 8 \cdot \frac{1}{2}t^{-\frac{1}{2}} = 4t^{-\frac{1}{2}} = \frac{4}{\sqrt{t}} \) - For **\( \mathbf{j} \)-component**: - The derivative of \( t^2 \sqrt{t} \): - \( t^2 \sqrt{t} = t^2 \cdot t^{\frac{1}{2}} = t^{2 + \frac{1}{2}} = t^{\frac{5}{2}} \) - The derivative of \( t^{\frac{5}{2}} \): - \( \frac{d}{dt}\left(t^{\frac{5}{2}}\right) = \frac{5}{2} t^{\frac{3}{2}} \) - For **\( \mathbf{k} \)-component**: - The derivative of \( \ln(t^2) \): - Using the property of logarithms, \( \ln(t^2) = 2 \ln(t) \) - The derivative of \(

Name: 12.2q3 ### Problem: Finding the Derivative of a Vector-Valued Function
Uploaded: 2024-03-20T06:46:14.000Z
Channel: Bartleby
Description: 12.2q3 ### Problem: Finding the Derivative of a Vector-Valued FunctionGiven the vector-valued function:\[ \mathbf{r}(t) = 8 \sqrt{t} \mathbf{i} + t^2 \sqrt{t} \mathbf{j} + \ln(t^2) \mathbf{k} \]We are tasked with finding the derivative \(\mathbf{r}'(