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)+y Now let's square both sides: y2=(x−sin‌x)+y Isolate y on one side:‌y2−y=x−sin‌x ‌y2−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 : ‌
d
dx
(y2−y−(x−sin‌x))=‌
d
dx
(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 ‌