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

Explanation

To differentiate the function \( y = \log(\log(\log(x^5))) \), we will follow the chain rule step by step.

### Step 1: Rewrite the function
We start by rewriting the function for clarity:
\[ y = \log(\log(\log(x^5))) \]

### Step 2: Differentiate using the chain rule
We will apply the chain rule multiple times since we have nested logarithmic functions.

1. **Outer function**: The outermost function is \( \log(u) \) where \( u = \log(\log(x^5)) \).
- The derivative of \( \log(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \).

2. **Middle function**: The middle function is \( v = \log(x^5) \).
- The derivative of \( \log(v) \) is \( \frac{1}{v} \cdot \frac{dv}{dx} \).

3. **Innermost function**: The innermost function is \( x^5 \).
- The derivative of \( x^5 \) is \( 5x^4 \).

### Step 3: Apply the chain rule
Now we differentiate step by step:

1. Differentiate the outer function:
\[ \frac{dy}{dx} = \frac{1}{\log(\log(x^5))} \cdot \frac{d}{dx}[\log(\log(x^5))] \]

2. Differentiate the middle function:
\[ \frac{d}{dx}[\log(\log(x^5))] = \frac{1}{\log(x^5)} \cdot \frac{d}{dx}[\log(x^5)] \]

3. Differentiate the innermost function:
\[ \frac{d}{dx}[\log(x^5)] = \frac{1}{x^5} \cdot 5x^4 = \frac{5}{x} \]

### Step 4: Combine the derivatives
Now we combine all the parts:
\[ \frac{dy}{dx} = \frac{1}{\log(\log(x^5))} \cdot \frac{1}{\log(x^5)} \cdot \frac{5}{x} \]

### Step 5: Simplify the expression
We can simplify the expression further:
\[ \frac{dy}{dx} = \frac{5}{x \cdot \log(x^5) \cdot \log(\log(x^5))} \]

### Final Result
Thus, the derivative of \( y = \log(\log(\log(x^5))) \) is:
\[ \frac{dy}{dx} = \frac{5}{x \cdot \log(x^5) \cdot \log(\log(x^5))} \]