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The problem asks for the derivative \( f'(x) \) of the function \( f(x) = 6x\sqrt{x} + \frac{6}{x^2\sqrt{x}} \), and then to evaluate \( f'(4) \).

### Step 1: Simplify the Function

Before differentiating, it will be easier to express the function in terms of powers of \( x \):

- Recall that \( \sqrt{x} = x^{1/2} \), so:
  \[
  6x\sqrt{x} = 6x \cdot x^{1/2} = 6x^{3/2}
  \]
- The second term is \( \frac{6}{x^2\sqrt{x}} \). Using \( \sqrt{x} = x^{1/2} \), we have:
  \[
  \frac{6}{x^2\sqrt{x}} = 6x^{-2} \cdot x^{-1/2} = 6x^{-5/2}
  \]
So the function becomes:
\[
f(x) = 6x^{3/2} + 6x^{-5/2}
\]

### Step 2: Differentiate \( f(x) \)

Now we differentiate term by term using the power rule:
- The derivative of \( 6x^{3/2} \) is:
  \[
  \frac{d}{dx}(6x^{3/2}) = 6 \cdot \frac{3}{2} x^{1/2} = 9x^{1/2}
  \]
- The derivative of \( 6x^{-5/2} \) is:
  \[
  \frac{d}{dx}(6x^{-5/2}) = 6 \cdot \frac{-5}{2} x^{-7/2} = -15x^{-7/2}
  \]

So, the derivative of \( f(x) \) is:
\[
f'(x) = 9x^{1/2} - 15x^{-7/2}
\]

### Step 3: Find \( f'(4) \)

Now we substitute \( x = 4 \) into the derivative expression:
\[
f'(4) = 9 \cdot 4^{1/2} - 15 \cdot 4^{-7/2}
\]

First, calculate the powers of 4:
- \( 4^{1/2} = \sqrt{4} = 2 \)
- \( 4^{-7/2} = \frac{1}{4^{7/2}} = \frac{1}{128} \)

Now, substitute these values:
\[
f'(4) = 9 \cdot 2 - 15 \cdot \frac{1}{128} = 18 - \frac{15}{128}
\]

Next, find a common denominator to simplify:
\[
18 = \frac{2304}{128}, \quad \text{so} \quad f'(4) = \frac{2304}{128} - \frac{15}{128} = \frac{2289}{128}
\]

Thus, \( f'(4) = \frac{2289}{128} \).

### Final Answer:
\( f'(4) = \frac{2289}{128} \).

Would you like more details or have any questions?

Here are five related questions:
1. How do you differentiate functions with fractional exponents?
2. What are some common applications of the power rule in real-world problems?
3. How does the chain rule apply to expressions involving square roots?
4. What is the difference between differentiating polynomials and rational functions?
5. How can you simplify complex functions before applying derivative rules?

**Tip:** Always simplify expressions before differentiating, as it makes the process easier and reduces the chance of errors.

If f(x) = 6x√x + 6/(x^2√x), then find f'(4).

Learn how to differentiate the function f(x) = 6x√x + 6/(x^2√x) and find the value of f'(4). This solution uses the power rule and involves simplifying expressions before differentiation. Suitable for high school students in Grades 11-12.

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