derivative of x*e^{1/x}

Lösung

\frac{d}{d x} (x \cdot e^{\frac{1}{x}})

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

Schritte zur Lösung

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

Wende die Produktregel an:

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

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

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

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

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

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

Vereinfache

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

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

x \to 1 lim (x (1))

\int_{4}^{8} \frac{2}{x ^{3}} d x

\int \frac{1}{24 x + 6 x ^{2}} d x

\int \frac{7}{7 x ^{2} + 14 x + 5} d x

x \to 0 lim (\frac{1}{x})