2 Prove by mathematical induction that, for all positive integers \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \tan ^ { - 1 } x \right) = P _ { n } ( x ) \left( 1 + x ^ { 2 } \right) ^ { - n } ,$$ where \(P _ { n } ( x )\) is a polynomial of degree \(n - 1\).
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