find-dy-dx-if-x-x-y-y-1-

Question Number 11545 by Nayon last updated on 28/Mar/17

f i n d \frac{d y}{d x} i f x^{x} y^{y} = 1

Answered by Joel576 last updated on 28/Mar/17

y^{y} = x^{- x}

y \ln y = - x \ln x

\frac{d}{d y} y \ln y = \frac{d}{d x} - x \ln x

\frac{d y}{d x} \ln y + 1 = - (\ln x + 1)

\frac{d y}{d x} = - \frac{\ln x + 1}{\ln y + 1}

Answered by ajfour last updated on 28/Mar/17

taking natural logarithm :

xln x + yln y = 0

differentiating we get,

\ln x + 1 + \frac{dy}{dx} \ln y + \frac{dy}{dx} = 0

\frac{dy}{dx} (1 + \ln y) = - (1 + \ln x)

\frac{dy}{dx} = - (\frac{1 + \ln x}{1 + \ln y}) = - \frac{\ln (ex)}{\ln (ey)} .

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