derivative of x^2arctan(7x)

解

\frac{d}{d x} (x^{2} arctan (7 x))

2 x arctan (7 x) + \frac{7 x ^{2}}{49 x ^{2} + 1}

解答ステップ

\frac{d}{d x} (x^{2} arctan (7 x))

積の法則を適用:

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

f = x^{2}, g = arctan (7 x)

= \frac{d}{d x} (x^{2}) arctan (7 x) + \frac{d}{d x} (arctan (7 x)) x^{2}

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

\frac{d}{d x} (arctan (7 x)) = \frac{7}{49 x ^{2} + 1}

= 2 x arctan (7 x) + \frac{7}{49 x ^{2} + 1} x^{2}

分数を乗じる:

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

= 2 x arctan (7 x) + \frac{7 x ^{2}}{49 x ^{2} + 1}