MHB Differentiation of infinite series

[Ande Yashwanth]

Find derivative of y=✓{x+✓[y+✓(x+...)]}infinite.
Here root comes for total inter terms

 

[HallsofIvy]

Find derivative of y=✓{x+✓[y+✓(x+...)]}infinite.
Here root comes for total inter terms

y= \sqrt{x+ \sqrt{y+ \sqrt{x+ \cdot\cdot\cdot}}} is clearly the same as y= \sqrt{x+ \sqrt{y+ y}}= \sqrt{x+ \sqrt{2y}}.

Squaring both sides, y^2= x+ \sqrt{2y} so that y^2- x= \sqrt{2y} and, squaring again, y^4- 2xy^2+ x^2= 2y or y^4- 2xy^2- 2y+ x^2= 0. Differentiating that with respect to x, 4y^3y'- 2y^2- 4xyy'- 2y'+ 2x= 0. Then 4y^3y'- 4xyy'- 2y'= (4y^2- 4xy- 2)y'= 2y^2- 2x so y'= \frac{y^2- x}{2y^2- 2xy- 1}.

(Shouldn't this be in "Calculus" rather than "Linear and Abstract Algebra"?)