If `y="log"_(2)"log"_(2)(x)`, then `(dy)/(dx)` is equal to

Options

A. `1/x log_(2) e. log_(x) e`
B. `1/x log_(2) x `
C. ` 1/x log_(e) x`
D. ` 1/x log_(x) e`

Answer

To solve the problem, we need to differentiate the function \( y = \log_2(\log_2(x)) \). Let's go through the steps one by one.

### Step 1: Rewrite the logarithm
Using the change of base formula for logarithms, we can express \( y \) in terms of natural logarithms:
\[ y = \log_2(\log_2(x)) = \frac{\log_e(\log_e(x))}{\log_e(2)} \]
Here, \( \log_e \) denotes the natural logarithm (base \( e \)).

### Step 2: Differentiate \( y \)
Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{\log_e(2)} \cdot \frac{d}{dx}[\log_e(\log_e(x))] \]

### Step 3: Differentiate \( \log_e(\log_e(x)) \)
Using the chain rule, we differentiate \( \log_e(\log_e(x)) \):
\[ \frac{d}{dx}[\log_e(\log_e(x))] = \frac{1}{\log_e(x)} \cdot \frac{d}{dx}[\log_e(x)] \]

### Step 4: Differentiate \( \log_e(x) \)
Now, we differentiate \( \log_e(x) \):
\[ \frac{d}{dx}[\log_e(x)] = \frac{1}{x} \]

### Step 5: Combine the results
Substituting back into our previous expression:
\[ \frac{d}{dx}[\log_e(\log_e(x))] = \frac{1}{\log_e(x)} \cdot \frac{1}{x} \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{1}{\log_e(2)} \cdot \left(\frac{1}{\log_e(x)} \cdot \frac{1}{x}\right) \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{1}{x \cdot \log_e(2) \cdot \log_e(x)} \]

### Final Answer
Therefore, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{1}{x \cdot \log_e(2) \cdot \log_e(x)} \]

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