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The image contains exercises focused on differentiation (derivatives). Here's a breakdown of the problems:

### EJERCICIO 1:
You are asked to calculate the derivative of the following functions:
- **a)** \( f(x) = x^3 + \frac{1}{x} \)
- **b)** \( g(x) = 3x^3 - \frac{x^2}{3} + \sqrt{2}x - \sqrt{3} \)
- **c)** \( h(x) = \frac{3x^5}{5} - \frac{2x^2}{3} + \frac{2}{3} \)
- **d)** \( j(x) = \frac{1}{2x} - \frac{2}{x^3} + \frac{\sqrt{3}}{\sqrt{2}} \)

### EJERCICIO 2:
Again, calculate the derivative of these functions:
- **a)** \( f(x) = x^4 \cdot \ln x \)
- **b)** \( g(x) = \sqrt{x} \cdot \cos x \)
- **c)** \( h(x) = \frac{3x}{3x - 1} \)
- **d)** \( j(x) = \frac{\sqrt{x}}{\sqrt{x + 1}} \)

### EJERCICIO 3:
You need to calculate the slope of the tangent line to the function \( f(x) \) from **Ejercicio 2a** at \( x = 1 \).

### EJERCICIO 4:
Calculate the slope of the tangent line to the function \( h(x) \) from **Ejercicio 2c** at \( x = 1 \).

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Would you like me to go through the detailed solution of each problem one by one, starting with **Ejercicio 1a**? 

Here are five related questions to deepen your understanding:
1. What are the general rules for calculating derivatives of sums and differences?
2. How do you apply the product rule for differentiation in **Ejercicio 2a**?
3. Can you explain the chain rule and where it is applied in **Ejercicio 2d**?
4. What is the significance of finding the slope of the tangent line at a specific point?
5. How can you simplify expressions like \( \frac{3x}{3x - 1} \) before differentiating?

**Tip**: When dealing with square roots in derivatives, rewriting them as powers (e.g., \( \sqrt{x} = x^{1/2} \)) simplifies the differentiation process.

Calculate the derivative of the given functions in exercises 1, 2, and the tangent lines in exercises 3 and 4.

Solve various differentiation problems, including derivatives of algebraic functions using power, product, and chain rules. Learn to calculate the slope of tangent lines at specific points.

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