derivative of 1/((a^2+x^2^{n/2)})

解

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

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

解答ステップ

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

指数の規則を適用する:

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

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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