derivative of x^3arctan(3x)

解

\frac{d}{d x} (x^{3} arctan (3 x))

3 x^{2} arctan (3 x) + \frac{3 x ^{3}}{9 x ^{2} + 1}

解答ステップ

\frac{d}{d x} (x^{3} arctan (3 x))

積の法則を適用:

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

f = x^{3}, g = arctan (3 x)

= \frac{d}{d x} (x^{3}) arctan (3 x) + \frac{d}{d x} (arctan (3 x)) x^{3}

\frac{d}{d x} (x^{3}) = 3 x^{2}

\frac{d}{d x} (arctan (3 x)) = \frac{3}{9 x ^{2} + 1}

= 3 x^{2} arctan (3 x) + \frac{3}{9 x ^{2} + 1} x^{3}

分数を乗じる:

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

= 3 x^{2} arctan (3 x) + \frac{3 x ^{3}}{9 x ^{2} + 1}