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Question Number 169612 by Mastermind last updated on 04/May/22

$$Differentiatew.r.t{}^{\prime }{x}^{\prime }$$$$x^y+y^x=c$$$$$$$$Mastermind$$

Answered by thfchristopher last updated on 04/May/22

$$\Rightarrow e^{\ln x^y}+e^{\ln y^x}=c$$$$\Rightarrow e^{y\ln x}+e^{x\ln y}=c$$$$\Rightarrow e^{y\ln x}(\frac{y}{x}+\ln x\frac{dy}{dx})+e^{x\ln y}(\ln y+\frac{x}{y}\cdot \frac{dy}{dx})=0$$$$\Rightarrow x^y\frac{y}{x}+x^y\ln x\frac{dy}{dx}+y^x\ln y+y^x\frac{x}{y}\cdot \frac{dy}{dx}=0$$$$\Rightarrow (x^y\ln x+{xy}^{x-1})\frac{dy}{dx}=-(x^{y-1}y+y^x\ln y)$$$$\frac{dy}{dx}=-\frac{x^{y-1}y+y^x\ln y}{x^y\ln x+{xy}^{x-1}}$$

Commented by Mastermind last updated on 04/May/22

$$Thanks$$

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