Question

If y=(tan-1x)^2 then prove that (1+x^2)^2 you + 2x ( 1+x^2) y1=2​

Answer 1

Given,

$\longrightarrow y=(\tan^{-1}x)^2$

Differentiating wrt $x,$

$\longrightarrow y'=((\tan^{-1}x)^2)'$

$\longrightarrow y'=\dfrac{2\tan^{-1}x}{1+x^2}$

$\longrightarrow (1+x^2)\,y'=2\tan^{-1}x$

Again differentiating wrt $x,$

$\longrightarrow ((1+x^2)\,y')'=(2\tan^{-1}x)'$

$\longrightarrow (1+x^2)\,y''+2xy'=\dfrac{2}{1+x^2}$

$\longrightarrow (1+x^2)\left[(1+x^2)\,y''+2xy'\right]=2$

$\longrightarrow\underline{\underline{(1+x^2)^2\,y''+2x(1+x^2)y'=2}}$

Hence Proved!