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Let's go through each question one by one.

### Pregunta 7:
You are asked to find the derivative of the function:

\[
y = \sqrt{3} + \cos(5x)
\]

The derivative of a constant \(\sqrt{3}\) is 0. The derivative of \(\cos(5x)\) is found using the chain rule:
\[
\frac{d}{dx}[\cos(5x)] = -\sin(5x) \cdot 5 = -5\sin(5x)
\]

Thus, the derivative of the function is:
\[
y' = -5\sin(5x)
\]

Now, look for the matching option in the provided choices.

### Pregunta 8:
The function is:
\[
f(x) = 3x^2 + 2x + 1
\]

We need to find the second derivative. First, let's find the first derivative:
\[
f'(x) = 6x + 2
\]

Now, the second derivative is:
\[
f''(x) = 6
\]

So the answer is **6**.

### Pregunta 9:
Given the equation:

\[
\cos(x)e^y - x^2 = 1
\]

We need to find \(\frac{dy}{dx}\). We'll use **implicit differentiation** for this problem.

1. Differentiate both sides of the equation with respect to \(x\):

   \[
   \frac{d}{dx} [\cos(x)e^y] - \frac{d}{dx}[x^2] = \frac{d}{dx}[1]
   \]

2. Using the **product rule** for \(\cos(x)e^y\):
   \[
   \frac{d}{dx}[\cos(x)e^y] = \sin(x) e^y + \cos(x) e^y \frac{dy}{dx}
   \]

3. The derivative of \(x^2\) is \(2x\). So, the equation becomes:
   \[
   -\sin(x) e^y + \cos(x) e^y \frac{dy}{dx} - 2x = 0
   \]

4. Solving for \(\frac{dy}{dx}\):
   \[
   \cos(x) e^y \frac{dy}{dx} = 2x + \sin(x) e^y
   \]
   \[
   \frac{dy}{dx} = \frac{2x + \sin(x) e^y}{\cos(x) e^y}
   \]

Now, match this with the given options.

---

Would you like further details or have any specific questions?

Here are 5 related questions for deeper understanding:
1. How do you apply the product rule in implicit differentiation?
2. What is the chain rule and how does it work for trigonometric functions?
3. Why is the derivative of a constant always zero?
4. What is the significance of the second derivative in calculus?
5. How do you differentiate composite functions?

**Tip**: Always check if implicit differentiation requires you to use both the product and chain rules when multiple variables are involved.

Mathematical problems involving derivatives and implicit differentiation.

Explore detailed solutions to calculus problems, including finding derivatives of trigonometric functions, using implicit differentiation, and calculating second derivatives. This guide is ideal for university-level students studying Calculus I.

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