D/dx(sin^4(cot^-1(1-x/1+x)^1/2)))?

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Answer 1 · 2017-11-25T18:14:35

$1/2(1+x).$

Given that, $y=sin^4(cot^-1sqrt{(1-x)/(1+x)}),$ we need $dy/dx.$

We substitute $x=cos2theta," so that, "-1 lt x lt 1.$

Note that, in order to make $sqrt((1-x)/(1+x))$ meaningful, we must have

$-1 lt x lt 1,$ which justifies our substitution : $x=cos2theta.$

$:. y=sin^4(cot^-1sqrt{(1-x)/(1+x)}),$

$=sin^4(cot^-1sqrt{(1-cos2theta)/(1+cos2theta)}),$

$=sin^4(cot^-1sqrt{(2sin^2theta)/(2cos^2theta)}),$

$=sin^4(cot^-1(tantheta)),$

$=sin^4(cot^-1{cot(pi/2-theta)}),$

$=sin^4(pi/2-theta),$

$={sin(pi/2-theta)}^4,$

$=cos^4theta,$

$=(cos^2theta)^2,$

$={(1+cos2theta)/2}^2.$

$rArr y=1/4(1+x)^2........................................[because, cos2theta=x].$

$:. dy/dx=1/4*d/dx(1+x)^2,$

$=1/4*2(1+x)*d/dx(1+x)..............[because," the Chain Rule],"$

$rArr dy/dx=1/2(1+x).$

Enjoy Maths.!