To find the derivative of the function y=√sin‌x+y, we will use implicit differentiation. Given the function: y=√sin‌x+y First, square both sides to eliminate the square root: y2=sin‌x+y Next, differentiate both sides with respect to x. Remember to use the chain rule and implicit differentiation for y : ‌
d
dx
(y2)=‌
d
dx
(sin‌x+y) Using the chain rule on the left side, we get: 2y‌
dy
dx
=cos‌x+‌
dy
dx
Now, isolate ‌
dy
dx
: ‌2y‌
dy
dx
−‌
dy
dx
=cos‌x ‌(2y−1)‌
dy
dx
=cos‌x ‌‌
dy
dx
=‌
cos‌x
2y−1
Now, substitute x=0 and y=1 into the equation: ‌
dy
dx
|x=0,y=1=‌
cos(0)
2⋅1−1
Since cos(0)=1 ‌
dy
dx
|x=0,y=1=‌
1
2−1
‌
dy
dx
|x=0,y=1=‌
1
1
So, the derivative ‌
dy
dx
at x=0 and y=1 is 1 . Hence, the correct answer is: Option B: 1