Q9 / 46 · NTA Abhyas (JEE Main) 2020 · Mathematics · Differentiation

If $l n (x + y) = 2 x y, then y^{'} (0)$ is equal to

Solution

Given equation is

\ln (x + y) = 2 x y

...(1)

For

x = 0

\ln (0 + y) = 2.0. y = 0

\Rightarrow \ln y = 0

\Rightarrow y = 1

Now, differentiating (1), we get,

\frac{1}{(x + y)} (1 + \frac{d y}{d x}) = 2 y + 2 x \frac{d y}{d x}

At point (0,1), we get,

(\frac{1}{0 + 1}) (1 + \frac{d y}{d x}) = 2 (1) + 2.0 . (\frac{d y}{d x})

\Rightarrow 1 + \frac{d y}{d x} = 2

\Rightarrow \frac{d y}{d x} = 1

= y^{'} (0)

Hence, the correct answer is option: 'A'.