The derivative of `y = sqrt (x +sqrt ( x + sqrt x))` has to be determined.

Use the chain rule.

y = `sqrt (x +sqrt ( x + sqrt x))`

y' = `(1 + (1/(2*sqrt ( x + sqrt x)))(1 + 1/(2*sqrt x)))/(2*sqrt(x + sqrt ( x + sqrt x)))`

=> `(1 + (1/(2*sqrt ( x + sqrt x)))((2*sqrt x +1)/(2*sqrt x)))/(2*sqrt(x + sqrt ( x + sqrt x)))`

=> `(1 + ((2*sqrt x + 1)/((sqrt ( x + sqrt x)*4*sqrt x))))/(2*sqrt(x + sqrt ( x + sqrt x)))`

=> `((4*sqrt x*sqrt(x+sqrt x) + 2*sqrt x + 1)/((sqrt ( x + sqrt x)*4*sqrt x)))/(2*sqrt(x + sqrt ( x + sqrt x)))`

=> `(4*sqrt x*sqrt(x+sqrt x) + 2*sqrt x + 1)/(8*sqrt x* sqrt(x + sqrt x)*sqrt(x + sqrt ( x + sqrt x)))`

**The derivative of ` y = sqrt (x +sqrt ( x + sqrt x))` is ** `y' =(4*sqrt x*sqrt(x+sqrt x) + 2*sqrt x + 1)/(8*sqrt x* sqrt(x + sqrt x)*sqrt(x + sqrt ( x + sqrt x)))`