Question

If \( y = \log_{2026}(\log_{2025} x) \), then \( \frac{dy}{dx} \)= _____

• A. \( \frac{1}{x \log x \log 2025} \)
• B. \( \frac{1}{x \log x \log 2026} \)
• C. \( \frac{1}{2025x \log x} \)
• D. \( \frac{1}{2026x \log x} \)

Hint:

Convert logarithms to natural log before differentiating.

Answer 1

The Correct Option is B

Concept: \[ \log_a x = \frac{\ln x}{\ln a} \]
Step 1: Rewrite function. \[ y = \frac{\ln(\log_{2025} x)}{\ln 2026} \]
Step 2: \[ \frac{dy}{dx} = \frac{1}{\ln 2026} \cdot \frac{1}{\log_{2025} x} \cdot \frac{d}{dx}(\log_{2025} x) \]
Step 3: \[ \frac{d}{dx}(\log_{2025} x) = \frac{1}{x \ln 2025} \]
Step 4: \[ \frac{dy}{dx} = \frac{1}{x \ln x \ln 2026} \]