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We are given the function \( f(x) = -\frac{2 \sqrt{x^3}}{5} + \frac{4}{x} \), and we are tasked with finding \( f'(4) \).

### Step 1: Rewrite the function
To make the differentiation easier, let's rewrite the function:
\[
f(x) = -\frac{2}{5} \cdot x^{3/2} + \frac{4}{x}
\]
Note that \( \sqrt{x^3} = x^{3/2} \).

### Step 2: Differentiate the function
We now differentiate \( f(x) \) term by term.

- For the first term \( -\frac{2}{5} \cdot x^{3/2} \), we use the power rule:
\[
\frac{d}{dx} \left( -\frac{2}{5} \cdot x^{3/2} \right) = -\frac{2}{5} \cdot \frac{3}{2} \cdot x^{1/2} = -\frac{3}{5} \cdot \sqrt{x}
\]

- For the second term \( \frac{4}{x} \), we can rewrite it as \( 4x^{-1} \), and differentiate:
\[
\frac{d}{dx} \left( 4x^{-1} \right) = -4x^{-2} = -\frac{4}{x^2}
\]

Thus, the derivative of \( f(x) \) is:
\[
f'(x) = -\frac{3}{5} \cdot \sqrt{x} - \frac{4}{x^2}
\]

### Step 3: Substitute \( x = 4 \) into the derivative
Now we find \( f'(4) \):
\[
f'(4) = -\frac{3}{5} \cdot \sqrt{4} - \frac{4}{4^2}
\]
\[
f'(4) = -\frac{3}{5} \cdot 2 - \frac{4}{16}
\]
\[
f'(4) = -\frac{6}{5} - \frac{1}{4}
\]

### Step 4: Combine the terms
To combine \( -\frac{6}{5} \) and \( -\frac{1}{4} \), we find a common denominator. The least common denominator of 5 and 4 is 20.

Convert both fractions:
\[
-\frac{6}{5} = -\frac{24}{20}, \quad -\frac{1}{4} = -\frac{5}{20}
\]

Now add them:
\[
f'(4) = -\frac{24}{20} - \frac{5}{20} = -\frac{29}{20}
\]

Thus, \( f'(4) = -\frac{29}{20} \).

### Final Answer:
\[
f'(4) = -\frac{29}{20}
\]

Would you like further details or clarifications? Here are some related questions:

1. How does the power rule help in differentiation?
2. Why is it important to simplify square roots in calculus problems?
3. What are common mistakes to avoid when applying the chain rule?
4. How can you verify your derivative calculations?
5. What are practical uses of derivatives in real-world scenarios?

**Tip:** Always simplify functions before differentiating to avoid unnecessary complexity.

Given the function \( f(x) = -\frac{2 \sqrt{x^3}}{5} + \frac{4}{x} \), find \( f'(4) \). Express your answer as a single fraction in simplest form.

Learn how to find the derivative of the function f(x) = -2√(x³)/5 + 4/x and evaluate it at x = 4. This step-by-step solution covers key calculus concepts, including the power rule for differentiation and algebraic manipulation. Suitable for students in Grades 11-12.

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