If x^y=y^x, then x(x-y x) \; dy/dx is equal to :

Correct Answer is : y(y−xlog⁡y)y(y - x y)y(y−xlogy)

Correct Answer is : y(y−xlog⁡y)y(y - x y)y(y−xlogy)

Test Index

EAMCET Engineering 2006 Solved Paper

Section: Mathematics

Question : 140 of 160

Marks: +1, -0

x^y=y^x

x(x-y \log x) \; \frac{dy}{dx}

Solution:

The given equation is

x^y=y^x

Taking log on the both sides, we get

\log (x^y) \; = \log (y^x)

\; \Rightarrow y \log x = x \log y

By differentiating both sides w.r.t

x

, we get

\log x \; \frac{dy}{dx} + y \left( \frac{1}{x} \right) \; = \log y(1) + x \left( \frac{1}{y} \right) \; \frac{dy}{dx}

\Rightarrow \;\; \log x \; \frac{dy}{dx} + \frac{y}{x} \; = \log y + \left( \frac{x}{y} \right) \; \frac{dy}{dx}

\Rightarrow \left( \frac{x}{y} - \log x \right) \; \frac{dy}{dx} \; = \frac{y}{x} - \log y

\; \Rightarrow \left( \frac{x - y \log x}{y} \right) \; \frac{dy}{dx} = \frac{y - x \log y}{x}

\; \Rightarrow x(x - y \log x) \; \frac{dy}{dx} = y(y - x \log y)

Go to Question:

Prev Question

Next Question