(\partial)/(\partial x)(e^{x/y}(1-x/y))

Lösung

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

- \frac{e ^{\frac{x}{y}} x}{y ^{2}}

Schritte zur Lösung

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

Behandle

y

als Konstante

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{\frac{x}{y}}, g = 1 - \frac{x}{y}

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

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

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

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

Vereinfache

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

= - \frac{e ^{\frac{x}{y}} x}{y ^{2}}

y^{^{'}} = \frac{x ^{2} + y ^{2}}{2 x y}

\frac{\partial}{\partial u ( v )} (\frac{e ^{v}}{u ( v ) + v ^{5}})

\frac{d}{d x} (\frac{x}{x ^{2} + 4})

x \to 25 lim (lo g_{5} (x))

\frac{d}{d x} (x^{5} ln (2 x))