Question

Find
$\frac{dy}{dx}$
​

Answer 1

y = √(sinx + √(sinx + √(sinx + ... ∞ ))) Square on both sides:

y² = sinx + √(sinx + √(sinx + ... ∞ )))

y² = sinx + y

y² - y = sinx

Differentiate both sides w.r.t x:

=> d(y² - y)/dx = d(sinx)/dx

=> d(y² - y)/dy × dy/dx = cosx

=> (2y - 1) × dy/dz = cosx

=> dy/dx = cosx/(2y - 1) proved

Line2: √(sinx + √(sinx + ... ∞ ))) is replaced with y as both are equal.

Line8: using chain rule

d(y² - y)/dx = d(y² - y)/dy * dy/dx