If y = log_{2026}(log_{2025} x), then frac{dy | vedclass

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Question

(C) Let $u = \log_{2025} x = \frac{\log x}{\log 2025}$.Then $y = \log_{2026} u = \frac{\log u}{\log 2026}$.Applying the chain rule:$\frac{dy}{dx} = \f

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$\frac{1}{x \log x \log 2026}$

Vedclass · Answer

(C) Let $u = \log_{2025} x = \frac{\log x}{\log 2025}$.Then $y = \log_{2026} u = \frac{\log u}{\log 2026}$.Applying the chain rule:$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \frac{1}{u \log 2026} \cdot \frac{d}{dx}\left(\frac{\log x}{\log 2025}\right)$.$\frac{dy}{dx} = \frac{1}{u \log 2026} \cdot \frac{1}{x \log 2025}$.Substituting $u = \frac{\log x}{\log 2025}$:$\frac{dy}{dx} = \frac{1}{(\frac{\log x}{\log 2025}) \log 2026 \cdot x \log 2025} = \frac{1}{x \log x \log 2026}$.

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

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