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Question

The differentiation of tan1(1+x21x) w.r.t. tan1x is

A
1
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B
12
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C
34
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D
12
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Solution

The correct option is B 12
Let u=tan1(1+x21x) and v=tan1xPutting x=tan θ, θ(π2,π2), we getu=tan1(1+tan2θ1tan θ)=tan1(sec θ1tan θ)u=tan1(1cos θsin θ)=tan1(2 sin2 θ22 sin θ2 cos θ2)u=tan1(tan θ2)=θ2=12tan1 xThus, we have u=12tan1 x and v=tan1 x.dudx=12×11+x2 and dvdx=11+x2dudv=du/dxdv/dx=12

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