InterviewSolution
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InterviewSolution
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If `y=(xsin^(-1)x)/(sqrt(1-x^2))+logsqrt(1-x^2)`, then prove that `(dy)/(dx)=(sin^(-1)x)/((1-x^2)^(3/2))` |
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Answer» Let `x = sin theta => dx/(d theta) = cos theta ` Now, `y = (theta sin theta)/sqrt(1-sin^2theta) + log sqrt(1-sin^2theta)` `=>y = (theta sin theta)/cos theta +log cos theta` `=>y = theta tan theta +log cos theta` `=>dy/(d theta) = tan theta + theta(sec^2 theta)+1/costheta(-sintheta)` `=>dy/(d theta) = tan theta + theta/(cos^2theta)-tan theta` `=>dy/(d theta) = theta/(cos^2theta)` `=> dy/dx = (dy/(d theta))/(dx/(d theta)) = (theta/(cos^2theta))/(cos theta)` `=> dy/dx = (theta/(cos^3 theta))` Now, `x = sin theta => cos theta = sqrt(1-x^2)` `=> dy/dx = (sin^-1x)/(1-x^2)^(3/2).` |
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