derivative of xarctan(2x-1/4 ln(1+4x^2))

解

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

arctan (2 x)

解答ステップ

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

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

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

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

簡素化

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

= arctan (2 x)