Question

Find the derivative of \[ \frac{d}{dx} \left( \log_3 x \cdot \log_x 3 \right). \]

• A. \( \frac{1}{9} \)
• B. 9
• C. \( 2 \log 3 \)
• D. 0

Hint:

Whenever you encounter a product of logarithms like this, simplify the expression before differentiating.

Answer 1

The Correct Option is D

We are asked to differentiate \( \log_3 x \cdot \log_x 3 \). Using properties of logarithms, we know that \( \log_3 x = \frac{\log x}{\log 3} \) and \( \log_x 3 = \frac{1}{\log x} \), so the expression simplifies to \( \frac{\log x}{\log 3} \cdot \frac{1}{\log x} = \frac{1}{\log 3} \). Differentiating this constant gives: \[ \frac{d}{dx} \left( \frac{1}{\log 3} \right) = 0. \] Thus, the correct answer is option (D) 0.