If $$\log(x + y) - 2xy = 0$$​, and $$y(0) = 1,$$​ then $$y'(0)$$  is:

Correct option is A

Given:
​$$\log(x + y) - 2xy = 0$$​​
Concept used:
Differentiate implicitly with respect to $$x.$$​​
Solution:
​$$\frac{d}{dx}[\log(x + y) - 2xy] = 0$$​​
​$$\Rightarrow \frac{1}{x + y}(1 + y') - 2(y + xy') = 0 \\\Rightarrow (1 + y') = 2(x + y)(y + xy') \\\Rightarrow 1 + y' = 2(xy + y^{2} + x^{2}y' + xy y')$$​​
At $$x = 0, y = 1:$$​​
​$$1 + y' = 2(0 + 1)(1 + 0 \cdot y') \\\Rightarrow 1 + y' = 2 \\\Rightarrow y' = 1$$​​
Correct answer is (a) 1