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

Answer

To find the derivative of \( y = \log(\log x) \), we will use the chain rule. Here’s the step-by-step solution:

### Step 1: Identify the outer and inner functions
We have:
- Outer function: \( f(u) = \log(u) \)
- Inner function: \( g(x) = \log(x) \)

So, we can express \( y \) as:
\[ y = f(g(x)) = \log(\log x) \]

### Step 2: Differentiate the outer function
Using the chain rule, we differentiate the outer function:
\[ \frac{dy}{du} = \frac{1}{u} \quad \text{where } u = g(x) = \log(x) \]

### Step 3: Differentiate the inner function
Next, we differentiate the inner function:
\[ \frac{du}{dx} = \frac{d}{dx}(\log x) = \frac{1}{x} \]

### Step 4: Apply the chain rule
Now, we apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{1}{\log x} \cdot \frac{1}{x} \]

### Step 5: Simplify the expression
Thus, we have:
\[ \frac{dy}{dx} = \frac{1}{x \log x} \]

### Final Answer
The derivative is:
\[ \frac{dy}{dx} = \frac{1}{x \log x} \]

---