derivative of arctan(x/b)

解

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

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

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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