derivative of xye^{-x^2-y^2}

Lösung

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

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

Schritte zur Lösung

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

Wende die Produktregel an:

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

f = x, g = y e^{- x^{2} - y^{2}}

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

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

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

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

Vereinfache

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

\frac{d}{d x} (e^{(- 4 x)})

y^{^{'}} + y = \frac{1}{1 + e ^{x}}

\int_{0}^{t} 5 e^{s} sin (t - s) d s

\int \frac{cos ( x )}{( sin ^{2} ( x ) ) ^{\frac{1}{3}}} d x

x \to 0 lim (\frac{sin ( 14 x )}{x})