Limit Continuity And Differentiability Ques 9
- If $y$ is a function of $x$ and $\log (x+y)=2 x y$, then the value of $y^{\prime}(0)$ is
(2004, 1M)
(a) $1$
(b) $-1$
(c) $2$
(d) $0$
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Answer:
Correct Answer: 9.(a)
Solution: (a) Given that, $\log (x+y)=2 x y$ $\quad$ ……..(i)
$\therefore \quad $ At $x=0, \Rightarrow \log (y)=0 \Rightarrow y=1$
$\therefore \quad $ To find $\frac{d y}{d x}$ at $(0,1)$
On differentiating Eq. (i) w.r.t. $x$, we get
$ \frac{1}{x+y}\left(1+\frac{d y}{d y}\right) =2 x \frac{d y}{d x}+2 y \cdot 1 $
$\Rightarrow \quad \frac{d y}{d x} =\frac{2 y(x+y)-1}{1-2(x+y) x} $
$\Rightarrow \quad \left(\frac{d y}{d x}\right)_{(0,1)} =1 $
BETA
