Question

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

(A) 1

(B) -1

(C) 2

(D) 0

Answer 1

SOLUTION — At $x=0$,

$\log (0+y)=2 \times 0 \times y$

$\Rightarrow y=1$

On differentiating given equation, we get

$\begin{aligned}\frac{1}{x+y}\left(1+\frac{d y}{d x}\right)  =2 x \frac{d y}{d x}+2 y \cdot 1 \\\Rightarrow \quad \frac{d y}{d x}=\frac{2 y(x+y)-1}{1-2(x+y) \cdot x} \Rightarrow\left(\frac{d y}{d x}\right)_{(0,1)} & =1\end{aligned}$

So, The correct option will be (A).