The derivative of tan-1x+1-x-1x+1+x-1 is
1-x2
11-x2
121-x2
x
Explanation for the correct option:
Finding the derivative of the given function:
Lety=tan-11+x-1-x1+x+1-x
Letx=cosθ⇒θ=cos-1x,then:
y=tan-11+cosθ-1-cosθ1+cosθ+1-cosθ
∵cos2θ=2cos2θ-1andcos2θ=1-2sin2θ
⇒cos2θ+1=2cos2θ⇒1-cos2θ=2sin2θ
Or,1+cosθ=2cos2θ2 Or,1-cosθ=2sin2θ2
∴y=tan-12cos2θ2-2sin2θ22cos2θ2+2sin2θ2=tan-12cosθ2-2sinθ22cosθ2+2sinθ2
=tan-12cosθ22cosθ2-2sinθ22cosθ22cosθ22cosθ2+2sinθ22cosθ2[dividingnumeratoranddenominatorby2cosθ2]
=tan-11-tanθ/21+tanθ/2=tan-1tanπ/4-tanθ/21+tanπ4tanθ2[∵tanπ4=1]
=tan-1tanπ4-θ2=π4-θ2=π4-cos-1x2
So,dydx=ddxπ4-cos-1x2=ddxπ4-12ddxcos-1x
=0-12×-11-x2
=121-x2
Therefore, the correct answer is option (C).