Let’s solve the given problem step by step.

Problem:

We are given $f(x) = \frac{4}{x} + 5x - 1$ and need to find $f'(2)$.

Solution:

1. Differentiate $f(x)$:

   Using the derivative rules:

   • The derivative of $\frac{4}{x} = 4x^{-1}$ is: $\frac{d}{dx} \left( \frac{4}{x} \right) = \frac{d}{dx} \left( 4x^{-1} \right) = -4x^{-2} = -\frac{4}{x^2}.$
   • The derivative of $5x$ is $5$.
   • The derivative of $-1$ is $0$.

   So, the derivative of $f(x)$ is: $f'(x) = -\frac{4}{x^2} + 5.$

2. Evaluate $f'(x)$ at $x = 2$: Substituting $x = 2$ into $f'(x) = -\frac{4}{x^2} + 5$: $f'(2) = -\frac{4}{2^2} + 5 = -\frac{4}{4} + 5 = -1 + 5 = 4.$

Final Answer:

$\boxed{4 \, \text{(A)}}$

Would you like me to explain any step further or explore related concepts?

Related Questions:

1. What are the derivative rules for other rational functions like $\frac{a}{x}$ or $\frac{a}{x^n}$?
2. How do we apply derivatives to solve real-world rate of change problems?
3. Can you find the second derivative $f''(x)$ for this function?
4. What happens to the value of $f'(x)$ as $x \to 0$?
5. How is $f(x)$ graphed, and how does the derivative relate to the slope?

Tip:

Always simplify the function (if possible) before taking its derivative to make the differentiation easier.