Question

find dy/dx,if y= sin^-1{(2√x)÷(1+x)}​

Answer 1

Given,

$\displaystyle\sf{\longrightarrow y=\sin^{-1}\left (\dfrac {2\sqrt x}{1+x}\right)\quad\quad\dots(1)}$

Substitute,

$\displaystyle\sf {\longrightarrow\sqrt x=\tan\theta\quad\quad\dots (2)}$

Differentiating,

$\displaystyle\sf {\longrightarrow\dfrac {1}{2\sqrt x}\ dx=\sec^2\theta\ d\theta}$

$\displaystyle\sf {\longrightarrow\dfrac {1}{2\sqrt x}\ dx=(1+\tan^2\theta)\ d\theta}$

From (2),

$\displaystyle\sf {\longrightarrow\dfrac {1}{2\sqrt x}\ dx=(1+x)\ d\theta}$

$\displaystyle\sf{\longrightarrow \dfrac {dx}{d\theta}=2(1+x)\sqrt x\quad\quad\dots(3)}$

Then (1) becomes,

$\displaystyle\sf{\longrightarrow y=\sin^{-1}\left (\dfrac {2\tan\theta}{1+\tan^2\theta}\right)}$

$\displaystyle\sf{\longrightarrow y=\sin^{-1}\left (\sin (2\theta)\right)}$

$\displaystyle\sf{\longrightarrow y=2\theta}$

Differentiating wrt $\displaystyle\sf {\theta,}$

$\displaystyle\sf{\longrightarrow \dfrac {dy}{d\theta}=2\quad\quad\dots(4)}$

Dividing (4) by (3),

$\displaystyle\sf{\longrightarrow \dfrac {\left (\dfrac {dy}{d\theta}\right)}{\left (\dfrac {dx}{d\theta}\right)}=\dfrac {2}{2(1+x)\sqrt x}}$

$\displaystyle\sf{\longrightarrow\underline {\underline {\dfrac {dy}{dx}=\dfrac {1}{(1+x)\sqrt x}}}}$

Done!