To solve this, we begin by finding the derivative of the inverse function $g(x)$. Here is a step-by-step explanation:

Problem Analysis:

The given function is: $f(x) = \ln(x^2) + 4x$ It has an inverse function $g(y)$, which satisfies: $g(f(x)) = x$ We need the slope of $g(x)$, which is determined by the derivative of $g(x)$: $g'(y) = \frac{1}{f'(g(y))}$

Step 1: Compute the derivative of $f(x)$

The function $f(x) = \ln(x^2) + 4x$ can be rewritten as: $f(x) = 2\ln|x| + 4x$ Differentiating with respect to $x$, we get: $f'(x) = \frac{2}{x} + 4$

Step 2: Express the slope of $g(x)$

The slope of the inverse function $g(y)$ is: $g'(y) = \frac{1}{f'(g(y))} = \frac{1}{\frac{2}{g(y)} + 4}$

Step 3: Substitution and Interpretation

To find the exact value for the slope $g'(y)$, you would need the specific value of $y$ or the range of $g(y)$. However, the formula for the slope is: $g'(y) = \frac{g(y)}{2 + 4g(y)}$

This provides the general slope for the inverse function.

Would you like further clarification or specific examples? Here are five follow-up questions for deeper understanding:

1. How do you compute the value of $g'(y)$ at a specific point $y$?
2. What is the significance of $g'(y)$ in terms of the behavior of the inverse function?
3. How does the domain of $f(x)$ affect the computation of $g'(y)$?
4. Can the function $g(y)$ be expressed explicitly?
5. What happens to $g'(y)$ as $y \to \infty$ or $y \to 0$?

Tip:

When dealing with inverse functions, remember that the derivative of the inverse can always be expressed as $g'(y) = \frac{1}{f'(g(y))}$. Use this formula to avoid directly solving for the inverse.