Differentiate `log[log(logx)]` w.r.t. x.

Answer

To differentiate the function \( y = \log(\log(\log x)) \) with respect to \( x \), we will use the chain rule. Here’s the step-by-step solution:

### Step 1: Identify the outer function
The outer function is \( f(u) = \log(u) \) where \( u = \log(\log x) \).

### Step 2: Differentiate the outer function
Using the derivative of the logarithm, we have:
\[ \frac{d}{du} f(u) = \frac{1}{u} = \frac{1}{\log(\log x)} \]

### Step 3: Identify the next inner function
Now, we need to differentiate \( g(v) = \log(v) \) where \( v = \log x \).

### Step 4: Differentiate the next inner function
Again, using the derivative of the logarithm:
\[ \frac{d}{dv} g(v) = \frac{1}{v} = \frac{1}{\log x} \]

### Step 5: Differentiate the innermost function
The innermost function is \( h(x) = \log x \). The derivative is:
\[ \frac{d}{dx} h(x) = \frac{1}{x} \]

### Step 6: Apply the chain rule
Now, we apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{1}{\log(\log x)} \cdot \frac{1}{\log x} \cdot \frac{1}{x} \]

### Step 7: Combine the results
Combining these results gives us:
\[ \frac{dy}{dx} = \frac{1}{x \log x \log(\log x)} \]

### Final Answer
Thus, the derivative of \( \log(\log(\log x)) \) with respect to \( x \) is:
\[ \frac{1}{x \log x \log(\log x)} \]

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