derivative 1/(d+ke^d)

解

導関数

\frac{1}{d + k e ^{d}}

- \frac{1 + k e ^{d}}{( d + k e ^{d} ) ^{2}}

解答ステップ

\frac{d}{dd} (\frac{1}{d + k e ^{d}})

指数の規則を適用する:

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

= \frac{d}{dd} ((d + k e^{d})^{- 1})

連鎖法則を適用:

- \frac{1}{( d + k e ^{d} ) ^{2}} \frac{d}{dd} (d + k e^{d})

= - \frac{1}{( d + k e ^{d} ) ^{2}} \frac{d}{dd} (d + k e^{d})

\frac{d}{dd} (d + k e^{d}) = 1 + k e^{d}

= - \frac{1}{( d + k e ^{d} ) ^{2}} (1 + k e^{d})

簡素化

- \frac{1}{( d + k e ^{d} ) ^{2}} (1 + k e^{d}) : - \frac{1 + k e ^{d}}{( d + k e ^{d} ) ^{2}}

= - \frac{1 + k e ^{d}}{( d + k e ^{d} ) ^{2}}