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To find the derivative \( f'(x) \) of the function \( f(x) = 6x^4 \sqrt{x} + \frac{-7}{x^2 \sqrt{x}} \), let's break down each term individually.

### Step 1: Simplify the Function
1. Rewrite \( \sqrt{x} \) as \( x^{1/2} \):
   \[
   f(x) = 6x^4 x^{1/2} - \frac{7}{x^2 x^{1/2}}
   \]

2. Use the laws of exponents to simplify:
   - For the first term \( 6x^4 x^{1/2} \): Combine the exponents:
     \[
     6x^4 x^{1/2} = 6x^{4 + 1/2} = 6x^{9/2}
     \]

   - For the second term \( \frac{-7}{x^2 x^{1/2}} \): Rewrite the denominator using exponents:
     \[
     \frac{-7}{x^2 x^{1/2}} = -7 x^{-(2 + 1/2)} = -7 x^{-5/2}
     \]

   So now, we have:
   \[
   f(x) = 6x^{9/2} - 7x^{-5/2}
   \]

### Step 2: Differentiate Each Term
Now that we have simplified \( f(x) \), we can differentiate each term separately using the power rule.

The power rule states that for \( f(x) = ax^n \), the derivative \( f'(x) = anx^{n-1} \).

1. Differentiate \( 6x^{9/2} \):
   \[
   \frac{d}{dx} \left( 6x^{9/2} \right) = 6 \cdot \frac{9}{2} x^{(9/2) - 1} = 27x^{7/2}
   \]

2. Differentiate \( -7x^{-5/2} \):
   \[
   \frac{d}{dx} \left( -7x^{-5/2} \right) = -7 \cdot \left( -\frac{5}{2} \right) x^{(-5/2) - 1} = \frac{35}{2} x^{-7/2}
   \]

### Step 3: Combine the Results
The derivative \( f'(x) \) is:
\[
f'(x) = 27x^{7/2} + \frac{35}{2}x^{-7/2}
\]

### Final Answer
\[
f'(x) = 27x^{7/2} + \frac{35}{2}x^{-7/2}
\]

Would you like more details or have any questions about this process?

### Related Questions
1. How do you apply the power rule to more complex expressions involving radicals?
2. What is the importance of simplifying expressions before differentiating?
3. How would you differentiate functions with higher-order roots, such as \( x^{1/3} \)?
4. What are some common mistakes when differentiating expressions with negative exponents?
5. How do you convert between radical and fractional exponent forms for differentiation?

### Tip
Always simplify your expressions as much as possible before taking derivatives; this makes it easier to apply rules correctly and avoid mistakes.

Let f(x) = 6x^4 √x + (-7) / (x^2 √x). Find f'(x).

Learn how to differentiate the function f(x) = 6x^4 √x + (-7) / (x^2 √x) by simplifying terms and applying the power rule. This solution includes step-by-step simplification and differentiation, making it suitable for students in Grades 11-12 studying calculus.

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