Differentiate `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 with the function:
\[ y = \log(\log(\log(x^5))) \]

### Step 2: Differentiate the outermost logarithm
Using the chain rule, the derivative of \( \log(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \). Here, let \( u = \log(\log(x^5)) \):
\[ \frac{dy}{dx} = \frac{1}{\log(\log(x^5))} \cdot \frac{d}{dx}[\log(\log(x^5))] \]

### Step 3: Differentiate the middle logarithm
Now we need to differentiate \( \log(\log(x^5)) \). Again, using the chain rule, let \( v = \log(x^5) \):
\[ \frac{d}{dx}[\log(\log(x^5))] = \frac{1}{\log(x^5)} \cdot \frac{d}{dx}[\log(x^5)] \]

### Step 4: Differentiate the innermost logarithm
Next, we differentiate \( \log(x^5) \). Using the property of logarithms \( \log(x^m) = m \log(x) \):
\[ \frac{d}{dx}[\log(x^5)] = \frac{d}{dx}[5 \log(x)] = 5 \cdot \frac{1}{x} = \frac{5}{x} \]

### Step 5: Combine the derivatives
Now we can substitute back into our previous differentiation:
\[ \frac{d}{dx}[\log(\log(x^5))] = \frac{1}{\log(x^5)} \cdot \frac{5}{x} \]

### Step 6: Substitute back into the derivative
Now substitute this back into the derivative of \( y \):
\[ \frac{dy}{dx} = \frac{1}{\log(\log(x^5))} \cdot \left(\frac{5}{x \log(x^5)}\right) \]

### Step 7: Simplify the expression
Now we can simplify the expression:
\[ \frac{dy}{dx} = \frac{5}{x \log(x^5) \log(\log(x^5))} \]

### Step 8: Substitute \( \log(x^5) \) back
Recall that \( \log(x^5) = 5 \log(x) \):
\[ \frac{dy}{dx} = \frac{5}{x (5 \log(x)) \log(\log(x^5))} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{1}{x \log(x) \log(\log(x^5))} \]

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

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