Given the function:

$f(x) = \frac{8}{\sqrt[4]{x}}$

To evaluate $f'(6)$, we'll first need to find the derivative of the function.

Step 1: Rewrite the function

The fourth root can be rewritten as an exponent. Thus:

$f(x) = 8x^{-\frac{1}{4}}$

Step 2: Differentiate the function

Now, apply the power rule for derivatives, which states:

$\frac{d}{dx}[x^n] = n x^{n-1}$

Differentiate $f(x) = 8x^{-\frac{1}{4}}$:

$f'(x) = 8 \times \left(-\frac{1}{4}\right) x^{-\frac{1}{4} - 1}$

Simplifying:

$f'(x) = -2 x^{-\frac{5}{4}}$

Step 3: Evaluate at $x = 6$

Substitute $x = 6$ into the derivative:

$f'(6) = -2 \times 6^{-\frac{5}{4}}$

We can compute this expression numerically.

Final Answer:

$f'(6) \approx -2 \times 6^{-1.25} \approx -2 \times 0.2902 \approx -0.5804$

So, $f'(6) \approx -0.5804$.

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Would you like further details on any part of this solution, or do you have any questions about derivatives or exponents?

Here are 5 related questions for you to explore:

1. What is the power rule for derivatives?
2. How do you differentiate functions with negative exponents?
3. Can you explain the fourth root in terms of exponents?
4. How does the chain rule work for functions like $f(x) = \frac{8}{\sqrt[4]{x}}$?
5. What is the significance of a derivative at a specific point, like $f'(6)$?

Tip: Practice differentiating simple power functions first (like $x^n$) before applying the power rule to more complex expressions!