derivative of xarctan(2x)

解

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

arctan (2 x) + \frac{2 x}{4 x ^{2} + 1}

解答ステップ

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

積の法則を適用:

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

f = x, g = arctan (2 x)

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

\frac{d x}{d x} = 1

\frac{d}{d x} (arctan (2 x)) = \frac{2}{4 x ^{2} + 1}

= 1 \cdot arctan (2 x) + \frac{2}{4 x ^{2} + 1} x

簡素化

1 \cdot arctan (2 x) + \frac{2}{4 x ^{2} + 1} x : arctan (2 x) + \frac{2 x}{4 x ^{2} + 1}

= arctan (2 x) + \frac{2 x}{4 x ^{2} + 1}