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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.

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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.

Differentiate the function f(t) = ³√t / (t - 9).

Learn how to differentiate the function f(t) = ³√t / (t - 9) step-by-step using the quotient rule. This problem is ideal for students in Grades 11-12 studying calculus, covering key concepts such as differentiation and the quotient rule.

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