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Question Number 26983 by chernoaguero@gmail.com last updated on 31/Dec/17
$$\mathrm{Find}\:\frac{\mathrm{dy}}{\mathrm{dx}}\:\:\:\: \\ $$$$\:\:\:\:\mathrm{x}^{\mathrm{y}} =\mathrm{y}^{\mathrm{x}} \\ $$
Answered by mrW1 last updated on 31/Dec/17
$${y}\:\mathrm{ln}\:{x}={x}\:\mathrm{ln}\:{y} \\ $$$${y}'\:\mathrm{ln}\:{x}+\frac{{y}}{{x}}=\mathrm{ln}\:{y}+\frac{{x}}{{y}}\:{y}' \\ $$$$\left(\frac{{x}}{{y}}−\mathrm{ln}\:{x}\right){y}'=\frac{{y}}{{x}}−\mathrm{ln}\:{y} \\ $$$$\frac{{dy}}{{dx}}={y}'=\frac{\frac{{y}}{{x}}−\mathrm{ln}\:{y}}{\frac{{x}}{{y}}−\mathrm{ln}\:{x}} \\ $$
Commented by chernoaguero@gmail.com last updated on 31/Dec/17
$$\mathrm{Good}\:\mathrm{thankx}\:\mathrm{u}\:\:\mathrm{sir} \\ $$