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CUET Mathematics 2026 PYQs26 May Shift 2. Free, no login required.

Calculus

past year questions, CUET Mathematics 2026 26 May Shift 2

Continuity & Differentiability

past year questions, Calculus, CUET Mathematics 2026 26 May Shift 2

Medium

core

If y=(log⁡ex)log⁡exy = (\log_e x)^{\log_e x}, then dydx\dfrac{dy}{dx} is equal to:

Solution

✅ Correct Option: 4

Take logarithms: log⁡y=log⁡ex⋅log⁡e(log⁡ex)\log y = \log_e x \cdot \log_e(\log_e x).

Differentiating, 1ydydx=1xlog⁡e(log⁡ex)+log⁡ex⋅1xlog⁡ex=1+log⁡e(log⁡ex)x\dfrac{1}{y}\dfrac{dy}{dx} = \dfrac{1}{x}\log_e(\log_e x) + \log_e x\cdot\dfrac{1}{x\log_e x} = \dfrac{1+\log_e(\log_e x)}{x}.

Hence dydx=(log⁡ex)log⁡ex[1+log⁡e(log⁡ex)x]\dfrac{dy}{dx} = (\log_e x)^{\log_e x}\left[\dfrac{1+\log_e(\log_e x)}{x}\right].

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