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Question: If y = \(x^{x^{x^{...\infty}}}\), then x \(\frac{dy}{dx}\) equals –...

If y = xxx...∞x^{x^{x^{...\infty}}}, then x dydx\frac{dy}{dx} equals –

A

x(1−ylog⁡x)y2\frac{x(1 - y\log x)}{y^{2}}

B

y2x(1−ylog⁡x)\frac{y^{2}}{x(1 - y\log x)}

C

y21−ylog⁡x\frac{y^{2}}{1 - y\log x}

D

None of these

Answer

y21−ylog⁡x\frac{y^{2}}{1 - y\log x}

Explanation

Solution

y = xy ⇒ log y = y log x.

∴  1ydydx\frac{1}{y}\frac{dy}{dx} = yx\frac{y}{x} + log x . dydx\frac{dy}{dx}

⇒ (1y−log⁡x)\left( \frac { 1 } { y } - \log x \right) dydx\frac{dy}{dx} = yx\frac{y}{x}

⇒ dydx\frac{dy}{dx} = yx\frac{y}{x}

⇒ x dydx\frac{dy}{dx} = y21−ylog⁡x\frac{y^{2}}{1 - y\log x}.