Differentiate: `log[log(logx^(5))]`

Explanation

To differentiate the function \( y = \log[\log(\log(x^5))] \), we will use the chain rule and properties of logarithms. Let's go through the steps one by one.

### Step 1: Rewrite the function
We start by rewriting the function using properties of logarithms:
\[ y = \log[\log(\log(x^5))] \]
Using the property of logarithms, we can express \( \log(x^5) \) as:
\[ \log(x^5) = 5 \log(x) \]
Thus, we can rewrite \( y \) as:
\[ y = \log[\log(5 \log(x))] \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we apply the chain rule. The derivative of \( \log(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \). Here, let:
\[ u = \log(5 \log(x)) \]
Then,
\[ y = \log(u) \]
Now, we differentiate:
\[ \frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx} \]

### Step 3: Differentiate \( u \)
Next, we need to differentiate \( u = \log(5 \log(x)) \). Again, using the chain rule:
\[ \frac{du}{dx} = \frac{1}{5 \log(x)} \cdot \frac{d}{dx}(5 \log(x)) \]
Since \( \frac{d}{dx}(\log(x)) = \frac{1}{x} \), we have:
\[ \frac{d}{dx}(5 \log(x)) = 5 \cdot \frac{1}{x} = \frac{5}{x} \]
Thus, substituting back, we get:
\[ \frac{du}{dx} = \frac{1}{5 \log(x)} \cdot \frac{5}{x} = \frac{1}{x \log(x)} \]

### Step 4: Substitute back into the derivative of \( y \)
Now substituting \( u \) and \( \frac{du}{dx} \) back into the derivative of \( y \):
\[ \frac{dy}{dx} = \frac{1}{\log(5 \log(x))} \cdot \frac{1}{x \log(x)} \]
Thus, we can express the final derivative as:
\[ \frac{dy}{dx} = \frac{1}{x \log(x) \log(5 \log(x))} \]

### Final Answer
The derivative of \( y = \log[\log(\log(x^5))] \) is:
\[ \frac{dy}{dx} = \frac{1}{x \log(x) \log(5 \log(x))} \]