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Beliebt Rechnen >

(\partial)/(\partial y)((e^x)/(1+e^y))

Lösung

\frac{\partial}{\partial y} (\frac{e ^{x}}{1 + e ^{y}})

Lösung

- \frac{e ^{x + y}}{( 1 + e ^{y} ) ^{2}}

Schritte zur Lösung

\frac{\partial}{\partial y} (\frac{e ^{x}}{1 + e ^{y}})

Behandle

x

als Konstante

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= e^{x} \frac{\partial}{\partial y} (\frac{1}{1 + e ^{y}})

Wende Exponentenregel an:

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

= e^{x} \frac{\partial}{\partial y} ((1 + e^{y})^{- 1})

Wende die Kettenregel an:

- \frac{1}{( 1 + e ^{y} ) ^{2}} \frac{\partial}{\partial y} (1 + e^{y})

= - \frac{1}{( 1 + e ^{y} ) ^{2}} \frac{\partial}{\partial y} (1 + e^{y})

\frac{\partial}{\partial y} (1 + e^{y}) = e^{y}

= e^{x} (- \frac{1}{( 1 + e ^{y} ) ^{2}} e^{y})

Vereinfache

e^{x} (- \frac{1}{( 1 + e ^{y} ) ^{2}} e^{y}) : - \frac{e ^{x + y}}{( 1 + e ^{y} ) ^{2}}

= - \frac{e ^{x + y}}{( 1 + e ^{y} ) ^{2}}

Beliebte Beispiele

derivative f(x)=3

d er i v a t i v e f (x) = 3

d/(dθ)((csc(θ)+cot(θ))^{-1})

\frac{d}{d θ} ((csc (θ) + cot (θ))^{- 1})

(dy)/(dx)+e^{x+y}=0

\frac{d y}{d x} + e^{x + y} = 0

integral of (x^2)/((x^3+5)^2)

\int \frac{x ^{2}}{( x ^{3} + 5 ) ^{2}} d x

(\partial)/(\partial x)(sqrt(4x+y^3))

\frac{\partial}{\partial x} (4 x + y^{3})