To find the derivative of the function \( F(x) = x^2 \sqrt{x} - 5x \), we can follow these steps:

### 1. Rewrite the function:
Since \( \sqrt{x} = x^{1/2} \), the function becomes:
\[
F(x) = x^2 x^{1/2} - 5x = x^{5/2} - 5x
\]

### 2. Differentiate each term:

- The derivative of \( x^{5/2} \) using the power rule \( \frac{d}{dx}[x^n] = n x^{n-1} \) is:
\[
\frac{d}{dx}[x^{5/2}] = \frac{5}{2} x^{3/2}
\]

- The derivative of \( -5x \) is:
\[
\frac{d}{dx}[-5x] = -5
\]

### 3. Combine the results:
So, the derivative of \( F(x) \) is:
\[
F'(x) = \frac{5}{2} x^{3/2} - 5
\]

This is the derivative of the given function.

Would you like further details or explanations on any of the steps? 

Here are 5 related questions to expand on this:
1. How does the power rule apply to fractional exponents?
2. What is the derivative of \( x^n \) when \( n \) is negative?
3. How do you find the derivative of a product of functions?
4. How do fractional powers of \( x \) relate to roots?
5. What is the geometric interpretation of this derivative?

**Tip:** When differentiating powers of \( x \), always reduce the exponent by 1 after multiplying by the original exponent.

Find the derivative of F(x) = x^2 sqrt(x) - 5x using the power rule and fractional exponents. Learn step-by-step how to rewrite the function and apply differentiation rules. This problem covers key calculus concepts suitable for Grades 10-12.

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