We are given the function $f(x) = -\frac{2\sqrt{x^3}}{5} - 3\sqrt{x}$, and we are tasked with finding the derivative $f'(x)$ and then evaluating $f'(5)$, expressing the answer in simplest radical form.

Step 1: Rewrite the function

First, simplify the function to make it easier to differentiate. Recall that: $\sqrt{x^3} = x^{3/2} \quad \text{and} \quad \sqrt{x} = x^{1/2}.$ So the function can be rewritten as: $f(x) = -\frac{2x^{3/2}}{5} - 3x^{1/2}.$

Step 2: Differentiate the function

Now, let's differentiate $f(x)$ with respect to $x$.

For the term $-\frac{2x^{3/2}}{5}$: $\frac{d}{dx}\left( -\frac{2x^{3/2}}{5} \right) = -\frac{2}{5} \cdot \frac{3}{2}x^{1/2} = -\frac{6}{5}x^{1/2}.$

For the term $-3x^{1/2}$: $\frac{d}{dx}\left( -3x^{1/2} \right) = -3 \cdot \frac{1}{2}x^{-1/2} = -\frac{3}{2}x^{-1/2}.$

Thus, the derivative is: $f'(x) = -\frac{6}{5}x^{1/2} - \frac{3}{2}x^{-1/2}.$

Step 3: Evaluate $f'(5)$

Now, substitute $x = 5$ into the derivative $f'(x)$: $f'(5) = -\frac{6}{5} \cdot \sqrt{5} - \frac{3}{2} \cdot \frac{1}{\sqrt{5}}.$

Simplify the expression: $f'(5) = -\frac{6\sqrt{5}}{5} - \frac{3}{2\sqrt{5}}.$

To combine these terms, express them with a common denominator. The least common denominator is $10\sqrt{5}$: $f'(5) = -\frac{12\sqrt{5}}{10} - \frac{3}{2\sqrt{5}} = -\frac{12\sqrt{5}}{10} - \frac{3\sqrt{5}}{10}.$

Now combine the terms: $f'(5) = -\frac{15\sqrt{5}}{10}.$

Simplify the fraction: $f'(5) = -\frac{3\sqrt{5}}{2}.$

Final Answer:

$f'(5) = -\frac{3\sqrt{5}}{2}.$

Would you like more details, or do you have any questions about the solution? Here are some related questions you might find useful:

1. How do you apply the chain rule when differentiating square roots of functions?
2. What is the derivative of $x^n$ in general, and how does it apply to this problem?
3. How do you handle fractional exponents when differentiating?
4. How can you simplify complex radical expressions after differentiation?
5. What strategies can be used to find common denominators when adding fractions with radicals?

Tip: Always simplify radicals and fractions after differentiation to make the final answer more concise.