derivative of-xe^{y/x}

Lösung

\frac{d}{d x} (- x e^{\frac{y}{x}})

- e^{\frac{y}{x}} - \frac{e ^{\frac{y}{x}} ( x \frac{d y}{d x} - y )}{x}

Schritte zur Lösung

\frac{d}{d x} (- x e^{\frac{y}{x}})

Entferne die Konstante:

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

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

Wende die Produktregel an:

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

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

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

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

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

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

Vereinfache

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

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

\int_{0}^{\frac{π}{2}} cot (x) d x

x \to 0 lim ((1 + 8 x)^{\frac{6}{x}})

s im pl i f y \frac{4 - 3 x - x ^{2}}{x ^{2} - 1}

\frac{d x}{d y} = 3 x

\int_{\frac{1}{2}}^{3} \frac{1}{x ^{2}} d x