InterviewSolution
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InterviewSolution
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Differentiate w.r.t. x: `(i)cos^(-1)((1-x^(2))/(1+x^(2)))" "(ii)sin^(-1)((2x)/(1+x^(2)))" "(iii)sec^(-1)((1)/(2x^(2)-1))` |
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Answer» (i) Let `y=cos^(-1)((1-x^(2))/(1+x^(2)))` Putting `x=tan theta,` we get `y=cos^(-1)((1-tan^(2)theta)/(1+tan^(2)theta))=cos^(-1)(cos2 theta)=2theta=2tan^(-1)x`. `therefore(dy)/(dx)=(2)/((1+x^(2))).` Hence, `(d)/(dx){cos^(-1)((1-x^(2))/(1+x^(2)))}=(2)/((1+x^(2))).` (ii) Let `y=sin^(-1)((2x)/(1+x^(2)))`. Putting `x=tan theta,` we get `y=sin^(-1)((2tan theta)/(1+tan^(2)theta))=sin^(-1)(sin 2 theta)=2 theta=2tan^(-1)x`. `therefore(dy)/(dx)=(2)/((1+x^(2)))`. Hence, `(d)/(dx){sin^(-1)((2x)/(1+x^(2)))}=(2)/((1+x^(2))).` (iii) Let `y=sec^(-1)((1)/(2x^(2)-1))`. Putting `x=cos theta,` we get `y=sec^(-1)((1)/(2cos^(2)theta-1))=sec^(-1)((1)/(cos 2 theta))` `=sec^(-1)(sec 2 theta)=2 theta=2cos^(-1)x`. `therefore y=2cos^(-1)x.` Hence, `(dy)/(dx)=2(d)/(dx)(cos^(-1)x)=(-2)/(sqrt(1-x^(2))).` |
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