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){(sqrt(1+tan^(2)theta-1))/(tan theta)}=tan^(-1){(sec theta-1)/(tan theta)}` `=tan^(-1){(((1)/(cos theta)-1))/(sin theta).cos theta}=tan^(=1)((1-cos theta)/(sin theta))` `=tan^(-1){(2sin^(2)(theta//2))/(2sin(theta//2)cos(theta//2))}=tan^(-1){"tan"(theta)/(2)}` `=(theta)/(2)=(1)/(2)tan^(-1)x.` `therefore y=(1)/(2)tan^(-1)x` `rArr(dy)/(dx)=(1)/(2).(d)/(dx)(tan^(-1)x)=(1)/(2(1+x^(2))).` |
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