InterviewSolution
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InterviewSolution
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If `y=sin^(-1){(sqrt(1+x)-sqrt(1-x))/(2)}`, find `(dy)/(dx).` |
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Answer» Putting `x=cos 2 theta,` we get `y=sin^(-1){(sqrt(1+cos 2theta)-sqrt(1-cos2 theta))/(2)}` `=sin^(-1){(sqrt(2cos^(2)theta)-sqrt(2sin^(2)theta))/(2)}=sin^(-1){(sqrt(2)cos^(2)theta-sqrt(2sin^(2)theta))/(2)}=sin^(-1){(sqrt2cos theta-sqrt2 sin theta)/(2)}` `=sin^(-1){(1)/(sqrt2)cos theta-(1)/(sqrt2)sin theta}=sin^(-1){"sin"(pi)/(4)cos theta-"cos"(pi)/(4)sin theta}` `=sin^(-1){sin((pi)/(4)-theta)}` `=((pi)/(4)-theta)=((pi)/(4)-(1)/(2)cos^(-1)x)` `" "[becausex=cos 2 thetarArr 2 theta=cos^(-1)x rArr theta=(1)/(2)cos^(-1)x]` `thereforey=(pi)/(4)-(1)/(2)cos^(-1)x.` Hence, `(dy)/(dx)=(d)/(dx){(pi)/(4)-(1)/(2)cos^(-1)x}=(d)/(dx)((pi)/(4))-(1)/(2)(d)/(dx)(cos^(-1)x)` `={0-(1)/(2).((-1))/(sqrt(1-x^(2)))}=(1)/(2sqrt(1-x^(2))).` |
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