let-y-x-x-x-2-calculate-dy-dx-

Question Number 42463 by abdo.msup.com last updated on 26/Aug/18

l e t y = \sqrt{x + \sqrt{x + \sqrt{x + 2}}}

c a l c u l a t e \frac{d y}{d x}

Commented by maxmathsup by imad last updated on 26/Aug/18

w e h a v e y^{2} = x + \sqrt{x + \sqrt{x + 2}} \Rightarrow {(y^{2} - x)}^{2} = x + \sqrt{x + 2} \Rightarrow

y^{4} - 2 {x y}^{2} + x^{2} = x + \sqrt{x + 2} l e t d e r i v a t e \Rightarrow

4 y^{^{'}} y^{3} - 2 (y^{2} + 2 {x y y}^{'}) + 2 x = 1 + \frac{1}{2 \sqrt{x + 2}} \Rightarrow

(4 y^{3} - 4 x y) y^{^{'}} - 2 y^{2} + 2 x = 1 + \frac{1}{2 \sqrt{x + 2}} \Rightarrow

(4 y^{3} - 4 x y) y^{^{'}} = 2 y^{2} - 2 x + 1 + \frac{1}{2 \sqrt{x + 2}} \Rightarrow

y^{^{'}} = \frac{2 y^{2} - 2 x + 1 + \frac{1}{2 \sqrt{x + 2}}}{4 y^{3} - 4 x y} .

Answered by tanmay.chaudhury50@gmail.com last updated on 26/Aug/18

y^{2} = x + \sqrt{x + \sqrt{x + 2}}

2 y \frac{d y}{d x} = (1 + \frac{1}{2 \sqrt{x + \sqrt{x + 2}}}) \times (1 + \frac{1}{2 \sqrt{x + 2}})

\frac{d y}{d x} = \frac{{1 + \frac{1}{2 (\sqrt{x + \sqrt{x + 2}}}} \times (1 + \frac{1}{2 \sqrt{x + 2}})}{2 y}