derivative x^2arctan(x)

解

導関数

x^{2} arctan (x)

2 x arctan (x) + \frac{x ^{2}}{x ^{2} + 1}

解答ステップ

\frac{d}{d x} (x^{2} arctan (x))

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{2}, g = arctan (x)

= \frac{d}{d x} (x^{2}) arctan (x) + \frac{d}{d x} (arctan (x)) x^{2}

\frac{d}{d x} (x^{2}) = 2 x

\frac{d}{d x} (arctan (x)) = \frac{1}{x ^{2} + 1}

= 2 x arctan (x) + \frac{1}{x ^{2} + 1} x^{2}

\frac{1}{x ^{2} + 1} x^{2} = \frac{x ^{2}}{x ^{2} + 1}

= 2 x arctan (x) + \frac{x ^{2}}{x ^{2} + 1}