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

1 Answer
Nov 25, 2017

# 1/2(1+x).#

Explanation:

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.!