Find `dy/dx if y=log(logx)`

Answer

To find \( \frac{dy}{dx} \) for the function \( y = \log(\log x) \), we will use the chain rule for differentiation. Here's a step-by-step solution:

### Step 1: Identify the function
We have:
\[ y = \log(\log x) \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we apply the chain rule. The chain rule states that if you have a composite function \( y = f(g(x)) \), then:
\[ \frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx} \]

In our case:
- Let \( g(x) = \log x \)
- Then \( y = \log(g(x)) \)

### Step 3: Differentiate the outer function
The derivative of \( y = \log(g) \) with respect to \( g \) is:
\[ \frac{dy}{dg} = \frac{1}{g} \]
Substituting \( g = \log x \):
\[ \frac{dy}{dg} = \frac{1}{\log x} \]

### Step 4: Differentiate the inner function
Next, we differentiate \( g(x) = \log x \) with respect to \( x \):
\[ \frac{dg}{dx} = \frac{1}{x} \]

### Step 5: Combine the derivatives
Now, we can combine the derivatives using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx} = \frac{1}{\log x} \cdot \frac{1}{x} \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{1}{x \log x} \]

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