Question

If $y = \log(\log x)$, then $dy/dx$ is:

• A. $\frac{1}{x \log x}$
• B. $\frac{1}{\log x}$
• C. $\frac{1}{x}$
• D. $\frac{\log x}{x}$

Hint:

Work from the outside in when using the chain rule.

Answer 1

The Correct Option is A

Step 1: Concept
Apply the Chain Rule: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$.
Step 2: Analysis
Let $u = \log x$. Then $y = \log u$. $dy/dx = (dy/du) \cdot (du/dx) = (1/u) \cdot (1/x)$.
Step 3: Conclusion
Substituting $u$ back: $dy/dx = \frac{1}{(\log x) \cdot x} = \frac{1}{x \log x}$.
Final Answer: (A)