derivative of arctan(1/2 x)

解

\frac{d}{d x} (arctan (\frac{1}{2} x))

\frac{2}{x ^{2} + 4}

解答ステップ

\frac{d}{d x} (arctan (\frac{1}{2} x))

簡素化

arctan (\frac{1}{2} x) : arctan (\frac{x}{2})

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

連鎖法則を適用:

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

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

\frac{d}{d x} (\frac{x}{2}) = \frac{1}{2}

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

簡素化

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

= \frac{2}{x ^{2} + 4}