Step 1: Understanding the Concept:
We use the change of base formula: $\log_a b = \frac{\ln b}{\ln a}$. This allows us to differentiate using the standard natural logarithm rules. Step 2: Formula Application:
Let $u = \log_{2025} x = \frac{\ln x}{\ln 2025}$.
Then $y = \log_{2026} u = \frac{\ln u}{\ln 2026}$.
By the Chain Rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. Step 3: Explanation:
$\frac{dy}{du} = \frac{1}{u \ln 2026}$ and $\frac{du}{dx} = \frac{1}{x \ln 2025}$.
Substituting $u$ back:
$$\frac{dy}{dx} = \frac{1}{\left(\frac{\ln x}{\ln 2025}\right) \ln 2026} \cdot \frac{1}{x \ln 2025} = \frac{1}{x \ln x \ln 2026}$$
(Note: $\log x$ in the options typically refers to $\ln x$ in calculus contexts). Step 4: Final Answer:
The correct option is (b).