InterviewSolution
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InterviewSolution
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Differentiate `tan^(-1)((sqrt(1+x^(2))+1)/(x))` w.r.t. x. |
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Answer» Let `y=tan^(-1)((sqrt(1+x^(2))+1)/(x)).` Putting `x=tan theta,` we get `y=tan^(-1)((sec theta+1)/(tan theta))=tan^(-1)((1+costheta)/(sin theta))` `=tan^(-1){(2cos^(2)(theta//2))/(2sin(theta//2)cos(theta//2))}=tan^(-1){cot.(theta)/(2)}` `=tan^(-1){tan((pi)/(2)-(theta)/(2))}=((pi)/(2)-(theta)/(2))=(pi)/(2)-(1)/(2)tan^(-1)x.` `therefore(dy)/(dx)=(1)/(2(1+x^(2))).` Hence, `(d)/(dx){tan^(-1)((sqrt(1+x^(2))+1)/(x))}=(-1)/(2(1+x^(2)))` |
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