derivative of ln((e^x/(1+e^x)))

Lösung

\frac{d}{d x} (ln (\frac{e ^{x}}{1 + e ^{x}}))

1 - \frac{e ^{x}}{1 + e ^{x}}

Schritte zur Lösung

\frac{d}{d x} (ln (\frac{e ^{x}}{1 + e ^{x}}))

Vereinfache

ln (\frac{e ^{x}}{1 + e ^{x}}) : x - ln (1 + e^{x})

= \frac{d}{d x} (x - ln (1 + e^{x}))

Wende die Summen-/Differenzregel an:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d x}{d x} - \frac{d}{d x} (ln (1 + e^{x}))

\frac{d x}{d x} = 1

\frac{d}{d x} (ln (1 + e^{x})) = \frac{e ^{x}}{1 + e ^{x}}

= 1 - \frac{e ^{x}}{1 + e ^{x}}

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