To differentiate the given vector function, we need to differentiate each component of the vector function $\mathbf{r}(t)$ separately.

The vector function is:

$\mathbf{r}(t) = \langle 8t e^{-t}, t \ln t, t \cos t \rangle$

Now, let's differentiate each component of $\mathbf{r}(t)$:

1. Differentiating the first component $8t e^{-t}$:

We will use the product rule since it is a product of two functions $8t$ and $e^{-t}$.

$\frac{d}{dt}(8t e^{-t}) = 8e^{-t} - 8t e^{-t} = 8e^{-t}(1 - t)$

2. Differentiating the second component $t \ln t$:

We will use the product rule for $t$ and $\ln t$.

$\frac{d}{dt}(t \ln t) = \ln t + 1$

3. Differentiating the third component $t \cos t$:

Again, use the product rule for $t$ and $\cos t$.

$\frac{d}{dt}(t \cos t) = \cos t - t \sin t$

Putting it all together:

The derivative of the vector function $\mathbf{r}(t)$ is:

$\mathbf{r}'(t) = \langle 8e^{-t}(1 - t), \ln t + 1, \cos t - t \sin t \rangle$

Let me know if you'd like further clarification on any part of the solution!

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Further Questions for Exploration:

1. How do you apply the product rule when differentiating functions like $t \ln t$ or $t \cos t$?
2. What is the general form of the product rule for differentiating products of functions?
3. How does the derivative of $e^{-t}$ differ from the derivative of $e^{t}$?
4. What happens to the derivative of each component as $t \to 0$ or as $t \to \infty$?
5. How do you differentiate more complex vector functions, such as those involving higher-order products or powers?

Tip: When applying the product rule, always differentiate each part of the product separately and then combine them using the rule: $(f \cdot g)' = f' \cdot g + f \cdot g'$.