The variables and (with and ) satisfy

MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

The variables xx and yy (with x>0x>0 and y>0y>0) satisfy xyyx=ex+yx^{y}y^{x}=e^{x+y}. Which of the following gives dydx\dfrac{dy}{dx}? (a) 1−yx+ln⁡yln⁡x+xy−1\displaystyle\frac{1-\dfrac{y}{x}+\ln y}{\ln x+\dfrac{x}{y}-1} (b) yx+ln⁡y−11−ln⁡x−xy\displaystyle\frac{\dfrac{y}{x}+\ln y-1}{1-\ln x-\dfrac{x}{y}} (c) 1−yx−ln⁡yln⁡x+xy−1\displaystyle\frac{1-\dfrac{y}{x}-\ln y}{\ln x+\dfrac{x}{y}-1} (d) 1−yx−ln⁡yln⁡x−1\displaystyle\frac{1-\dfrac{y}{x}-\ln y}{\ln x-1}

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