To solve this problem, let's first observe the nested square root structure within the equation
y=√(x−sinx)+√(x−sinx)+√(x−sinx)....., Since the structure of nested square roots repeats indefinitely, we can rewrite this as
y=√(x−sinx)+yNow let's square both sides:
y2=(x−sinx)+yIsolate
y on one side:
y2−y=x−sinxy2−y−(x−sinx)=0 This equation gives us a relationship between
x and
y that we can differentiate with respect to
x. We will use implicit differentiation, differentiating both sides of the equation with respect to
x :
(y2−y−(x−sinx))=(0)When differentiating the left side, keep in mind that
y is a function of
x (
y=f(x) ). Applying the chain rule and using the fact that
(sinx)=cosx, we get:
2y−−(1−cosx)=0 Reorganize the terms:
2y−=1−cosx(2y−1)=1−cosxNow, solving for
:
=This matches Option A: