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"?)