To find the derivative of the function \( f(q) = \left( \frac{64}{q} \right)^{\frac{1}{4}} \), we will proceed step by step:

### Step 1: Rewrite the function in a simpler form

We can express the function as:
\[
f(q) = \left( \frac{64}{q} \right)^{\frac{1}{4}} = 64^{\frac{1}{4}} \cdot q^{-\frac{1}{4}}
\]
Since \( 64^{\frac{1}{4}} = 2 \) (because \( 64 = 2^6 \), and \( \frac{1}{4} \) of 6 is 1.5, which gives 2), the function simplifies to:
\[
f(q) = 2 \cdot q^{-\frac{1}{4}}
\]

### Step 2: Differentiate the function

Now, take the derivative of \( f(q) = 2 \cdot q^{-\frac{1}{4}} \) with respect to \( q \):

Using the power rule \( \frac{d}{dq} q^n = n \cdot q^{n-1} \), we get:
\[
f'(q) = 2 \cdot \left(-\frac{1}{4}\right) \cdot q^{-\frac{1}{4} - 1}
\]

Simplifying this:
\[
f'(q) = -\frac{1}{2} \cdot q^{-\frac{5}{4}}
\]

### Final Answer:
The derivative of \( \left( \frac{64}{q} \right)^{\frac{1}{4}} \) is:
\[
f'(q) = -\frac{1}{2} \cdot q^{-\frac{5}{4}}
\]

Do you want further details or have any questions? Here are five related questions to consider:

1. What is the significance of negative exponents in derivatives?
2. How does the chain rule apply when differentiating more complex expressions?
3. Can you generalize the derivative of \( \left( \frac{A}{q} \right)^n \) for any constants \( A \) and \( n \)?
4. What are some practical applications of derivatives in real-life problems?
5. How would the derivative change if \( q \) was a function of another variable?

**Tip:** Always simplify functions before differentiating, as this makes the process easier and reduces errors.

Discover how to find the derivative of the function (64/q)^(1/4) using the power rule. This detailed solution includes exponent simplification and derivative steps, perfect for college-level calculus students.

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