derivative of arctan(sqrt(x/a))

解

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

\frac{a}{2 x ( x + a )}

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

= \frac{a}{2 x ( x + a )}