derivative y= 1/(sqrt(a^2-x^2))

解

導関数

y = \frac{1}{a ^{2} - x ^{2}}

\frac{x}{( a ^{2} - x ^{2} ) ^{\frac{3}{2}}}

解答ステップ

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

連鎖法則を適用:

- \frac{1}{2 ( a ^{2} - x ^{2} ) ^{\frac{3}{2}}} \frac{d}{d x} (a^{2} - x^{2})

= - \frac{1}{2 ( a ^{2} - x ^{2} ) ^{\frac{3}{2}}} \frac{d}{d x} (a^{2} - x^{2})

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

= - \frac{1}{2 ( a ^{2} - x ^{2} ) ^{\frac{3}{2}}} (- 2 x)

簡素化

- \frac{1}{2 ( a ^{2} - x ^{2} ) ^{\frac{3}{2}}} (- 2 x) : \frac{x}{( a ^{2} - x ^{2} ) ^{\frac{3}{2}}}

= \frac{x}{( a ^{2} - x ^{2} ) ^{\frac{3}{2}}}