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.