Question

The derivative of \( 2^x \) w.r.t. \( 3^x \) is:

• A. \( \frac{x}{2} \frac{\log 2}{\log 3} \)
• B. \( \frac{2x}{3} \frac{\log 2}{\log 3} \)
• C. \( \frac{x \log 2}{x \log 3} \)
• D. \( \frac{x \log 3}{x \log 2} \)

Hint:

For derivatives of exponential functions, use the chain rule along with logarithmic properties.

Answer 1

The Correct Option is C

The derivative of \( 2^x \) is: \[ \frac{d}{dx} (2^x) = 2^x \log 2. \] The derivative of \( 3^x \) is: \[ \frac{d}{dx} (3^x) = 3^x \log 3. \] Thus, the derivative of \( 2^x \) with respect to \( 3^x \) is: \[ \frac{\frac{d}{dx} (2^x)}{\frac{d}{dx} (3^x)} = \frac{2^x \log 2}{3^x \log 3}. \]
Final Answer: \( \boxed{ {(C)}} \)