The problem asks to find the derivative of the function:

$f(x) = \ln\left(\frac{x^3 e^{x^9}}{x^9 + 8}\right)$

We can solve this using the chain rule and logarithmic differentiation. Let’s break this down step by step.

Step 1: Use the logarithmic property for differentiation

The logarithmic rule says that:

$\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$

So, we can rewrite $f(x)$ as:

$f(x) = \ln(x^3 e^{x^9}) - \ln(x^9 + 8)$

Step 2: Simplify the expression

Now simplify the logarithms:

$\ln(x^3 e^{x^9}) = \ln(x^3) + \ln(e^{x^9}) = 3\ln(x) + x^9$

Thus, the function becomes:

$f(x) = 3\ln(x) + x^9 - \ln(x^9 + 8)$

Step 3: Differentiate the expression

Now differentiate term by term.

1. The derivative of $3\ln(x)$ is: $\frac{d}{dx} [3\ln(x)] = \frac{3}{x}$

2. The derivative of $x^9$ is: $\frac{d}{dx} [x^9] = 9x^8$

3. For $-\ln(x^9 + 8)$, we use the chain rule: $\frac{d}{dx} [-\ln(x^9 + 8)] = -\frac{1}{x^9 + 8} \cdot \frac{d}{dx}(x^9 + 8) = -\frac{9x^8}{x^9 + 8}$

Step 4: Combine the derivatives

Now, summing all the parts together, we get:

$f'(x) = \frac{3}{x} + 9x^8 - \frac{9x^8}{x^9 + 8}$

Step 5: Match the expression to the answer choices

Looking at the provided options, the correct choice is:

$\boxed{C}: \frac{3}{x} - 1 - \frac{9x^8}{x^9 + 8}$

Would you like more details on any step?

Here are 5 related questions to deepen your understanding:

1. What is the chain rule, and how is it used in this derivative?
2. Why do we use logarithmic properties to simplify complex expressions?
3. Can you explain how to differentiate $\ln(x)$?
4. How does the quotient rule compare to logarithmic differentiation?
5. How does the derivative of a product involving exponents, like $e^{x^9}$, differ from simpler products?

Tip: Whenever differentiating logarithmic expressions, simplify as much as possible using logarithmic identities before differentiating!