Question

If $\log (x+y)=2xy$, then $y^{\prime }\left(0\right)$ is equal to

• A. $1$
• B. $-1$
• C. $2$
• D. $0$

Answer 1

The correct option is A $1$

$\log (x+y)=2xy$
On differentiating both sides w.r.t $x$, we get
$\left(\frac{1}{x+y}\right)(1+\frac{dy}{dx})=2(x\frac{dy}{dx}+y)$
$\Rightarrow \frac{dy}{dx}=\frac{1-2xy-2y^2}{2x^2+2xy-1}$
at $x=0,y=1$ (from $\log x+y=2xy$)
Now, at $x=0,y^{\prime }\left(0\right)=\frac{1-2}{-1}=1$