Differentiate tan-1 [ (root 1 - x2) / x ] w.r.t. cos-1 [2x (root 1 - x2)].

Let u = tan-11-x2xput x = cos θ   u = tan-11-cos2θcosθu = tan-1sin θcos θu = tan-1tan θu = θu = cos-1xdudx = -11-x2

Let v = cos-12x1-x2put x = sin θso, v = cos-12 sin θ 1 - sin2θv = cos-12 sin θ cos θv = cos-1sin 2θv = cos-1cosπ2-2θv = π2-2θv = π2-2sin-1xv = -21-x2Now,dudv = dudx ×dxdv = -11-x2×-1-x22 = 12

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