Question

If $y$ is a function of $x$ and $\log (x+y)=2xy,$ then the value of $y(0)$ is equal to

• A. 1
• B. -1
• C. 2
• D. 0

Answer 1

Answer: (A)

Since to find $\frac{dy}{dx}$ at $x=0$
$\therefore$ At $x=0$
$\Rightarrow \log (y)=0$
$\Rightarrow y=1$
$\therefore$ To find $\frac{dy}{dx}$ at $(0,1)$.
On differentiating, $\log (x+y)=2xy$ on both sides we get
$\frac{1}{x+y}\left(1+\frac{dy}{dx}\right)=2x\frac{dy}{dx}+2y.1$
$\Rightarrow \frac{dy}{dx}=\frac{2y(x+y)-1}{1-2(x+y)x}$
$\therefore {\left(\frac{dy}{dx}\right)}_{(0,1)}=1$