`y =[ log_(x) (log _(e) x ) ](log _(e) x )` then `(dy)/(dx)` equals

Options

A. `(1)/( x log _(x) log _(x) x)`
B. `(1)/(x log _(e) x )`
C. 0
D. None of these

Answer

To find the derivative of the function \( y = \left[ \log_{x}(\log_{e} x) \right] (\log_{e} x) \), we will follow these steps:

### Step 1: Rewrite the function using properties of logarithms
We know that:
\[ \log_{a}(b) = \frac{\log_{e}(b)}{\log_{e}(a)} \]
Thus, we can rewrite \( \log_{x}(\log_{e} x) \) as:
\[ \log_{x}(\log_{e} x) = \frac{\log_{e}(\log_{e} x)}{\log_{e}(x)} \]
Now substituting this back into the equation for \( y \):
\[ y = \left( \frac{\log_{e}(\log_{e} x)}{\log_{e}(x)} \right) (\log_{e} x) \]
The \( \log_{e}(x) \) terms will cancel out:
\[ y = \log_{e}(\log_{e} x) \]

### Step 2: Differentiate using the chain rule
Now we need to differentiate \( y \):
\[ y = \log_{e}(\log_{e} x) \]
Using the chain rule, the derivative of \( \log_{e}(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \). Here, \( u = \log_{e} x \).

First, differentiate the outer function:
\[ \frac{dy}{du} = \frac{1}{\log_{e} x} \]

Now differentiate the inner function \( u = \log_{e} x \):
\[ \frac{du}{dx} = \frac{1}{x} \]

### Step 3: Combine the derivatives
Now we can combine these results:
\[ \frac{dy}{dx} = \frac{1}{\log_{e} x} \cdot \frac{1}{x} \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{1}{x \log_{e} x} \]

### Final Result
The derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{1}{x \log_{e} x} \]

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