Question

Differentiate \(\log(\log x)\), \(x>1\) with respect to \(x\).

Hint:

Chain rule: d/dx[log(log x)] = (1/log x)·(1/x).

Answer 1

Step 1: Key Formula or Approach:
Use the chain rule: if \(y=\log(u)\) where \(u=\log x\), then \(\dfrac{dy}{dx} = \dfrac1u\cdot\dfrac{du}{dx}\).

Step 2: Differentiating the outer function:
\[ \frac{d}{dx}\log(\log x) = \frac{1}{\log x}\cdot\frac{d}{dx}(\log x) \]

Step 3: Differentiating the inner function:
\(\dfrac{d}{dx}(\log x) = \dfrac1x\). Substituting:
\[ \frac{d}{dx}\log(\log x) = \frac{1}{\log x}\cdot\frac1x = \frac{1}{x\log x} \]

Final Answer:
The derivative is \(\dfrac{1}{x\log x}\). \[ \boxed{\dfrac{1}{x\log x}} \]