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