if-y-x-x-2-1-then-find-d-y-d-x-

Question Number 107922 by mohammad17 last updated on 13/Aug/20

i f : y = \frac{x}{x^{2} + 1} t h e n f i n d \frac{d \sqrt{y}}{d \sqrt{x}} ?

Answered by 1549442205PVT last updated on 13/Aug/20

Set \sqrt{x} = t and \sqrt{y} = z \Rightarrow y = \frac{t^{2}}{t^{4} + 1}

z^{2} = \frac{t^{2}}{t^{4} + 1} \Rightarrow 2 z . z^{'} = \frac{2 t (t^{4} + 1) - {4 t}^{3} . t^{2}}{{(t^{4} + 1)}^{2}}

\Rightarrow 2 z . z^{'} = \frac{- {2 t}^{5} + 2}{{(t^{4} + 1)}^{2}}

\frac{d \sqrt{y}}{d \sqrt{x}} = \frac{dz}{dt} = \frac{- {2 t}^{5} + 2}{{(t^{4} + 1)}^{2} . 2 z} = \frac{- {2 x}^{2} \sqrt{x} + 2}{2 {(x^{2} + 1)}^{2} \sqrt{\frac{x}{x^{2} + 1}}}

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