The function given in the image is:

$f(t) = \frac{\sqrt[3]{t}}{t - 3}$

We are asked to differentiate this function with respect to $t$.

Step-by-Step Differentiation

To differentiate this, we apply the quotient rule. The quotient rule states that if:

$f(t) = \frac{g(t)}{h(t)}$

then the derivative $f'(t)$ is:

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

Here, $g(t) = \sqrt[3]{t} = t^{1/3}$ and $h(t) = t - 3$.

1. Differentiate $g(t) = t^{1/3}$

Using the power rule:

$g'(t) = \frac{1}{3}t^{-\frac{2}{3}}$

2. Differentiate $h(t) = t - 3$

Since $h(t) = t - 3$, the derivative is simply:

$h'(t) = 1$

3. Apply the quotient rule

Now we substitute into the quotient rule:

$f'(t) = \frac{\left( \frac{1}{3}t^{-\frac{2}{3}} \right)(t - 3) - \left(t^{\frac{1}{3}}\right)(1)}{(t - 3)^2}$

4. Simplify the expression

Expanding the numerator:

$f'(t) = \frac{\frac{1}{3}t^{-\frac{2}{3}}(t - 3) - t^{\frac{1}{3}}}{(t - 3)^2}$

This is the derivative of the function. If you need further simplifications or details about this derivative, let me know!

Would you like more details on the simplification, or do you have any questions?

Here are 5 related questions for further exploration:

1. How does the power rule apply to functions of fractional exponents?
2. What are some other functions where the quotient rule is necessary?
3. How do you simplify expressions with negative exponents?
4. Can we apply the chain rule to more complex functions with cube roots?
5. How does the derivative behave near singularities like $t = 3$?

Tip: When using the quotient rule, always remember to check whether the denominator has any points of discontinuity (like division by zero).