derivative 1/(x^2-2x-3)

解

導関数

\frac{1}{x ^{2} - 2 x - 3}

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

解答ステップ

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

指数の規則を適用する:

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

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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