A) \[y/(x(1-2x))\]
B) \[x/(y\,(1-2x))\]
C) \[x/(y(1-2y))\]
D) \[y/(x(1-2y))\]
Correct Answer: D
Solution :
We have \[y=x{{e}^{2y}}\] Taking log on both sides, we get \[\log y=\log (x{{e}^{2y}})\] \[\Rightarrow \] \[\log y=\log x+2y\log e\] \[\Rightarrow \] \[\log y=\log x+2y\] On differentiating w. r. t. x, we get \[\frac{1}{y}\frac{dy}{dx}=\frac{1}{x}+2\frac{dy}{dx}\] \[\Rightarrow \] \[\frac{dy}{dx}\left( \frac{1}{y}-2 \right)=\frac{1}{x}\] \[\Rightarrow \] \[\frac{dy}{dx}=\frac{1}{x}\times \frac{y}{(1-2y)}\] \[\Rightarrow \] \[\frac{dy}{dx}=\frac{y}{x(1-2y)}\]
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