The problem asks to find the derivative of the function $y = \log |\log x|$. We assume that $\log$ denotes the natural logarithm, i.e., the logarithm with base $e$.

1. Problem Description

The problem asks to find the derivative of the function $y = \log |\log x|$. We assume that $\log$ denotes the natural logarithm, i.e., the logarithm with base $e$.

2. Solution Steps

We need to find $\frac{dy}{dx}$ for $y = \log |\log x|$.

Let $u = \log x$, so $y = \log |u|$.

Then, by the chain rule:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

First, we find $\frac{dy}{du}$.

Since $y = \log |u|$, we have

$\frac{dy}{du} = \frac{1}{u}$

Next, we find $\frac{du}{dx}$.

Since $u = \log x$, we have

$\frac{du}{dx} = \frac{1}{x}$

Now, we can find $\frac{dy}{dx}$:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \frac{1}{u} \cdot \frac{1}{x} = \frac{1}{\log x} \cdot \frac{1}{x} = \frac{1}{x \log x}$

Therefore, $\frac{dy}{dx} = \frac{1}{x \log x}$.

3. Final Answer

$\frac{dy}{dx} = \frac{1}{x \log x}$