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To solve the derivative of the given function:

\[
y = \ln\left(\left(\frac{x^2 + 2}{x^3}\right)^5\right)
\]

We will follow these steps:

1. **Apply the chain rule**: The logarithmic function and the power function are involved.
2. **Simplify the expression**: Use logarithmic properties to make the differentiation easier.
3. **Differentiate step by step**.

### Step 1: Simplify the expression using logarithm properties.
We can simplify the logarithm of a power:

\[
y = 5 \cdot \ln\left(\frac{x^2 + 2}{x^3}\right)
\]

Now, use the property of logarithms: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\):

\[
y = 5 \cdot \left(\ln(x^2 + 2) - \ln(x^3)\right)
\]

This further simplifies to:

\[
y = 5 \cdot \left(\ln(x^2 + 2) - 3\ln(x)\right)
\]

### Step 2: Differentiate with respect to \(x\).
Now, differentiate term by term.

- Derivative of \(5 \cdot \ln(x^2 + 2)\):
  - Use the chain rule: \( \frac{d}{dx}[\ln(u)] = \frac{1}{u} \cdot \frac{du}{dx} \).
  - Let \( u = x^2 + 2 \), then \( \frac{du}{dx} = 2x \).
  - So, the derivative is \( \frac{1}{x^2 + 2} \cdot 2x = \frac{2x}{x^2 + 2} \).

- Derivative of \(5 \cdot (-3 \ln(x))\):
  - Use the derivative of \( \ln(x) \), which is \( \frac{1}{x} \).
  - So, the derivative is \( -15 \cdot \frac{1}{x} = -\frac{15}{x} \).

### Step 3: Combine the results.
The derivative \( y' \) is:

\[
y' = 5 \cdot \left(\frac{2x}{x^2 + 2} - \frac{3}{x}\right)
\]

This is the derivative of the given function.

Would you like further details or clarifications on this solution?

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Here are 5 related questions:
1. How do you apply the chain rule in differentiation?
2. What are the properties of logarithms used in differentiation?
3. Can you differentiate more complex logarithmic functions?
4. How does the chain rule apply to nested functions?
5. How would the derivative change if the power were different, e.g., \( (x^2 + 2)^3 \)?

**Tip**: Simplifying expressions before differentiating often makes the process easier and reduces the chance of errors.

y = ln((x^2 + 2) / x^3)^5. Realiza las siguientes derivadas.

Learn how to find the derivative of the function y = ln((x^2 + 2) / x^3)^5 using logarithmic differentiation and the chain rule. This problem covers advanced calculus concepts such as logarithmic properties and the application of the chain rule.

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