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