InterviewSolution
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InterviewSolution
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Differentiate `tan^(-1)((sqrt(1+x^(2))-1)/(x))` w.r.t. `tan^(-1)x.` |
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Answer» Let `u=tan^(-1)((sqrt(1+x^(2))-1)/(x)) and v=tan^(-1)x` Now, `v=tan^(-1)x rArr x = tanv.` Putting `x=tan v,` we get `u=tan^(-1){(sqrt(1+tan^(2)v)-1)/(tanv)}=tan^(-1)((secv-1)/(tanv))` `=tan^(-1)((1-cosv)/(sinv))=tan^(-1){(2sin^(2)(v//2))/(2sin(v//2)cos(v//2))}` `=tan^(-1){tan.(v)/(2)}=(v)/(2).` `thereforeu=(v)/(2)rArr(du)/(dv)=(1)/(2).` |
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