1.

Find the differential coefficient of `(sin x -x cos x)/(x sin x+cos x)` with respect to x.

Answer» Let `y=(sin x-x cos x)/(x sin x+cos x)`
`(dy)/(dx)=(d)/(dx)((sinx-x cos x)/(x sinx+cos x))`
`(xsinx+cosx).(d)/(dx)(sinx-xcos x)`
`=(-(sinx-xcos x)(d)/(dx)(xsin+cosx))/((x sin x+cos x)^(2))`
`(xsinx+cos x)(cos x-cos x+x sinx)`
`=(-(sinx-xcoss)(x cos x+sinx-sinx))/((x sinx+cos x)^(2))`
`=((x sin x+cos x).x sin x-(sinx-x cos x)x cos x)/((x sin x+cos x)^(2))`
`=(x(xsin^(2)x+sinxcos x-sinxcos x+x cos^(2)x))/((x sin x+cos x)^(2))`
`=(x^(2)(sin^(2)x+cos^(2)x))/((x sin x+cos x)^(2))`
`(x^(2))/((x sin x +cos x)^(2))`


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