Differentiate $tan^{-1} \bigg[ \large\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}} \bigg]$ w.r.t. $sin^{-1} \bigg( \large\frac{2x}{1+x^2} \bigg).$

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