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)

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