derivative of (x^2+x^3e^{-1/x})

Lösung

\frac{d}{d x} ((x^{2} + x^{3}) e^{- \frac{1}{x}})

3 e^{- \frac{1}{x}} x^{2} + 3 e^{- \frac{1}{x}} x + e^{- \frac{1}{x}}

Schritte zur Lösung

\frac{d}{d x} ((x^{2} + x^{3}) e^{- \frac{1}{x}})

Wende die Produktregel an:

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

f = x^{2} + x^{3}, g = e^{- \frac{1}{x}}

= \frac{d}{d x} (x^{2} + x^{3}) e^{- \frac{1}{x}} + \frac{d}{d x} (e^{- \frac{1}{x}}) (x^{2} + x^{3})

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

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

= (2 x + 3 x^{2}) e^{- \frac{1}{x}} + \frac{e ^{- \frac{1}{x}}}{x ^{2}} (x^{2} + x^{3})

Vereinfache

(2 x + 3 x^{2}) e^{- \frac{1}{x}} + \frac{e ^{- \frac{1}{x}}}{x ^{2}} (x^{2} + x^{3}) : 3 e^{- \frac{1}{x}} x^{2} + 3 e^{- \frac{1}{x}} x + e^{- \frac{1}{x}}

= 3 e^{- \frac{1}{x}} x^{2} + 3 e^{- \frac{1}{x}} x + e^{- \frac{1}{x}}

\int_{0}^{π} 2 sin (x) + sin (2 x) d x

\int csc (π x) cos (π x) d x

\int cos (nπ t) d t

d er i v a t i v e 11 x \cdot e^{x}

x \to 0 lim (tanh (x))