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

Options

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

Explanation

To find the derivative of the function \( y = \log_{2}(\log_{2}(x)) \), we will use the chain rule and the properties of logarithms.

**Step 1: Rewrite the function using natural logarithms.**

We know that:
\[ \log_{a}(b) = \frac{\ln(b)}{\ln(a)} \]
Thus, we can rewrite \( y \) as:
\[ y = \frac{\ln(\log_{2}(x))}{\ln(2)} \]

**Step 2: Differentiate \( y \) with respect to \( x \).**

Using the chain rule:
\[ \frac{dy}{dx} = \frac{1}{\ln(2)} \cdot \frac{d}{dx}(\ln(\log_{2}(x))) \]

Now, we need to differentiate \( \ln(\log_{2}(x)) \). Again using the chain rule:
\[ \frac{d}{dx}(\ln(\log_{2}(x))) = \frac{1}{\log_{2}(x)} \cdot \frac{d}{dx}(\log_{2}(x)) \]

**Step 3: Differentiate \( \log_{2}(x) \).**

Using the change of base formula again:
\[ \log_{2}(x) = \frac{\ln(x)}{\ln(2)} \]
Thus,
\[ \frac{d}{dx}(\log_{2}(x)) = \frac{1}{\ln(2)} \cdot \frac{1}{x} \]

**Step 4: Substitute back into the derivative.**

Now substituting this back:
\[ \frac{d}{dx}(\ln(\log_{2}(x))) = \frac{1}{\log_{2}(x)} \cdot \frac{1}{\ln(2)} \cdot \frac{1}{x} \]

So, we have:
\[ \frac{dy}{dx} = \frac{1}{\ln(2)} \cdot \left( \frac{1}{\log_{2}(x)} \cdot \frac{1}{\ln(2)} \cdot \frac{1}{x} \right) \]
\[ = \frac{1}{\ln(2)^{2}} \cdot \frac{1}{x \cdot \log_{2}(x)} \]

**Final Answer:**
\[ \frac{dy}{dx} = \frac{1}{x \cdot \log_{2}(x) \cdot \ln(2)^{2}} \]

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