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

Answer: (D)

$\frac {y} {x(1-2y)}$

Answer 2

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 $\frac{1}{y}\frac{dy}{dx}=\frac{1}{x}+2\frac{dy}{dx}$
$\Rightarrow$ $\frac{dy}{dx}\left( \frac{1}{y}-2 \right)=\frac{1}{x}$
$\Rightarrow$ $\frac{dy}{dx}=\frac{1}{x}\times \frac{y}{(1-2y)}$
$\Rightarrow$ $\frac{dy}{dx}=\frac{y}{x(1-2y)}$