Question

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

• A. 1
• B. - 1
• C. 2
• D. 0

Answer 1

Answer: (A)

Given that, $\log (x + y) = 2 xy ....$(i)
$\therefore $ At $x = 0, \Rightarrow \log \, (y) = 0 \Rightarrow y = 1 $
$\therefore $ To find $ \frac{dy}{dx} $ at $(0, 1)$
On differentiating Eq. (i) w.r.t. $x$, we get
$ \frac{1}{ x + y } \bigg(1 + \frac{dy}{dx}\bigg) = 2x \frac{dy}{dx} + 2y . 1 $
$\Rightarrow \frac{dy}{dx} = \frac{ 2 y \, (x + y) - 1}{ 1 - 2 \, (x + y) \, x}$.
$\Rightarrow \left(\frac{dy}{dx}\right)_{(0, 1)} = 1$