derivative of a/(x^2+a^2)

解

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

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

解答ステップ

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

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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