derivative x^2arctan(5x)

解

導関数

x^{2} arctan (5 x)

2 x arctan (5 x) + \frac{5 x ^{2}}{25 x ^{2} + 1}

解答ステップ

\frac{d}{d x} (x^{2} arctan (5 x))

積の法則を適用:

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

f = x^{2}, g = arctan (5 x)

= \frac{d}{d x} (x^{2}) arctan (5 x) + \frac{d}{d x} (arctan (5 x)) x^{2}

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

\frac{d}{d x} (arctan (5 x)) = \frac{5}{25 x ^{2} + 1}

= 2 x arctan (5 x) + \frac{5}{25 x ^{2} + 1} x^{2}

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 2 x arctan (5 x) + \frac{5 x ^{2}}{25 x ^{2} + 1}