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The problem requires calculating the derivatives of the given functions and determining if the provided derivative expressions are true or false.

### Problem A:
Given the function:
\[ f(x) = e^x - \frac{1}{x} - 2 \]

The claim is that its derivative is:
\[ f'(x) = e^x + \frac{1}{x^2} \]

Let's calculate the derivative of \( f(x) \):
1. The derivative of \( e^x \) is \( e^x \).
2. The derivative of \( -\frac{1}{x} \) is \( \frac{1}{x^2} \) (since \( -\frac{1}{x} = -x^{-1} \)).
3. The derivative of the constant \( -2 \) is \( 0 \).

Thus, the derivative is:
\[ f'(x) = e^x + \frac{1}{x^2} \]

**Conclusion:** The given statement for part A is **true**.

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### Problem B:
Given the function:
\[ f(x) = x^{\frac{1}{2}} + \frac{2}{x} - 1 \]

The claim is that its derivative is:
\[ f'(x) = \frac{1}{2}x^{-\frac{1}{2}} - \frac{2}{x^2} \]

Let's calculate the derivative of \( f(x) \):
1. The derivative of \( x^{\frac{1}{2}} \) is \( \frac{1}{2}x^{-\frac{1}{2}} \) (using the power rule).
2. The derivative of \( \frac{2}{x} \) is \( -\frac{2}{x^2} \) (since \( \frac{2}{x} = 2x^{-1} \)).
3. The derivative of the constant \( -1 \) is \( 0 \).

Thus, the derivative is:
\[ f'(x) = \frac{1}{2}x^{-\frac{1}{2}} - \frac{2}{x^2} \]

**Conclusion:** The given statement for part B is also **true**.

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Would you like any further details or clarifications? Here are 5 related questions for further practice:

1. What is the derivative of \( f(x) = e^x - \frac{2}{x^3} \)?
2. How do you apply the power rule for derivatives?
3. What is the second derivative of \( f(x) = x^{\frac{1}{2}} \)?
4. How does the chain rule apply to derivatives of composite functions?
5. Can you explain the geometric interpretation of the derivative?

**Tip:** Always simplify the function as much as possible before differentiating, especially when working with negative exponents or fractions!

Realizar la derivada de las siguientes funciones. Indicar, justificando sus respuestas si los siguientes enunciados son verdaderos o falsos:
A) f(x) = e^x - 1/x - 2 → su derivada es f'(x) = e^x + 1/x^2
B) f(x) = x^(1/2) + 2/x - 1 → su derivada es f'(x) = 1/2 * x^(-1/2) - 2/x^2

This page explains how to calculate and verify the derivatives of functions involving exponents and fractions. It covers the application of the power rule, chain rule, and derivatives of exponential functions. Example problems include verifying the correctness of the derivative expressions for f(x) = e^x - 1/x - 2 and f(x) = x^(1/2) + 2/x - 1.

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