To differentiate the function $f(t) = \frac{2t}{4 + t^2}$, we will use the quotient rule, which is applied when differentiating a ratio of two functions.

The quotient rule states that if $f(t) = \frac{g(t)}{h(t)}$, then:

$f'(t) = \frac{g'(t)h(t) - g(t)h'(t)}{[h(t)]^2}$

Here, $g(t) = 2t$ and $h(t) = 4 + t^2$.

Step-by-step solution:

1. Differentiate the numerator $g(t) = 2t$: $g'(t) = 2$

2. Differentiate the denominator $h(t) = 4 + t^2$: $h'(t) = 2t$

3. Apply the quotient rule: $f'(t) = \frac{(2)(4 + t^2) - (2t)(2t)}{(4 + t^2)^2}$

4. Simplify the numerator:

   = \frac{8 + 2t^2 - 4t^2}{(4 + t^2)^2} = \frac{8 - 2t^2}{(4 + t^2)^2}$$

So, the derivative is:

$f'(t) = \frac{8 - 2t^2}{(4 + t^2)^2}$

Would you like further clarification or details on this solution?

Here are 5 questions that expand on this information:

1. How does the quotient rule compare with the product rule in terms of usage and application?
2. Can the derivative of this function be simplified further or factored?
3. How does the second derivative of this function behave, and what does it tell us about the function?
4. What would the graph of this function and its derivative look like?
5. Can you find a real-world example where this type of function might be used?

Tip:

When applying the quotient rule, always double-check the signs and terms in the numerator after differentiation to ensure correct simplification.