The problem asks to differentiate the function $f(t) = \frac{\sqrt[3]{t}}{t - 9}$.

To differentiate this, we need to apply the quotient rule, which is used for functions of the form $\frac{u(t)}{v(t)}$. The quotient rule states:

$\frac{d}{dt}\left( \frac{u(t)}{v(t)} \right) = \frac{u'(t)v(t) - u(t)v'(t)}{[v(t)]^2}$

Here:

• $u(t) = \sqrt[3]{t} = t^{1/3}$
• $v(t) = t - 9$

Step 1: Find the derivatives of $u(t)$ and $v(t)$

• $u'(t) = \frac{d}{dt}(t^{1/3}) = \frac{1}{3}t^{-2/3}$
• $v'(t) = \frac{d}{dt}(t - 9) = 1$

Step 2: Apply the quotient rule

Now, applying the quotient rule:

$f'(t) = \frac{\left( \frac{1}{3}t^{-2/3} \right)(t - 9) - (t^{1/3})(1)}{(t - 9)^2}$

Step 3: Simplify

Simplify the expression step by step: $f'(t) = \frac{\frac{1}{3}t^{-2/3}(t - 9) - t^{1/3}}{(t - 9)^2}$

This is the derivative of the given function.

Would you like a more detailed explanation or any further clarification? Here are 5 follow-up questions you might find useful:

1. How do you simplify expressions involving fractional exponents?
2. Can you use other methods to differentiate this function, like logarithmic differentiation?
3. What is the physical interpretation of a derivative in real-world applications?
4. How does the quotient rule relate to the product rule?
5. What are some common mistakes when applying the quotient rule?

Tip: Always simplify the individual derivatives before applying the quotient rule to avoid algebraic errors.