Question

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

• A. $1$
• B. $-1$
• C. $2$
• D. $0$

Answer 1

The correct option is A $1$

When $x=0$, we have

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

$\Rightarrow \log y=0$

$\Rightarrow y=e^0=1$

Given equation is

$\log (x+y)-2xy=0$

On differentiating w,r,t $x$ we get

$\frac{1}{x+y}(1+\frac{dy}{dx})-2y-2x\frac{dy}{dx}=0$

On putting $x=0,y=1$ we get

$(1+\frac{dy}{dx})-2-0=0$

$\Rightarrow (1+\frac{dy}{dx})-2=0$

$\Rightarrow \frac{dy}{dx}=1$

$ie,y^{\prime }\left(x\right)=1$

$\Rightarrow y^{\prime }\left(0\right)=1$