If y= ` e^(log (log x )) ,then (dy)/(dx) ` =

Options

A. ` (1)/(x) `
B. ` (1)/(logx ) `
C. ` (1)/( xlogx ) `
D. ` (x)/( log x ) `

Answer

To find the derivative of the function \( y = e^{\log(\log x)} \), we can follow these steps:

### Step 1: Simplify the Expression
Using the property of logarithms, we can rewrite the expression:
\[ y = e^{\log(\log x)} = \log x^{\log e} \]
Since \( \log e = 1 \), we have:
\[ y = \log x \]

### Step 2: Differentiate the Function
Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(\log x) \]
The derivative of \( \log x \) is:
\[ \frac{dy}{dx} = \frac{1}{x} \]

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

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