Question

If $ y=x{{e}^{2y}}, $ then find $\frac{ dy}{dx} $ .

• A. $\frac { y} {(x(1-2x))} $
• B. $\frac { x} {(y\,(1-2x))} $
• C. $\frac {x} {y(1-2y)} $
• D. $\frac {y} {x(1-2y)} $

Answer 1

The Correct Option is D

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)} $