d/(dy)((2-y^2+x^2)e^{(1-x^2-(y^2)/4)})

Lösung

\frac{d}{d y} ((2 - y^{2} + x^{2}) e^{(1 - x^{2} - \frac{y ^{2}}{4})})

- 2 e^{1 - x^{2} - \frac{y ^{2}}{4}} y - \frac{e ^{\frac{4 - 4 x ^{2} - y ^{2}}{4}} y ( 2 - y ^{2} + x ^{2} )}{2}

Schritte zur Lösung

\frac{d}{d y} ((2 - y^{2} + x^{2}) e^{(1 - x^{2} - \frac{y ^{2}}{4})})

Behandle

x

als Konstante

Wende die Produktregel an:

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

f = 2 - y^{2} + x^{2}, g = e^{1 - x^{2} - \frac{y ^{2}}{4}}

= \frac{d}{d y} (2 - y^{2} + x^{2}) e^{1 - x^{2} - \frac{y ^{2}}{4}} + \frac{d}{d y} (e^{1 - x^{2} - \frac{y ^{2}}{4}}) (2 - y^{2} + x^{2})

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

\frac{d}{d y} (e^{1 - x^{2} - \frac{y ^{2}}{4}}) = - \frac{e ^{\frac{4 - 4 x ^{2} - y ^{2}}{4}} y}{2}

= (- 2 y) e^{1 - x^{2} - \frac{y ^{2}}{4}} + (- \frac{e ^{\frac{4 - 4 x ^{2} - y ^{2}}{4}} y}{2}) (2 - y^{2} + x^{2})

Vereinfache

(- 2 y) e^{1 - x^{2} - \frac{y ^{2}}{4}} + (- \frac{e ^{\frac{4 - 4 x ^{2} - y ^{2}}{4}} y}{2}) (2 - y^{2} + x^{2}) : - 2 e^{1 - x^{2} - \frac{y ^{2}}{4}} y - \frac{e ^{\frac{4 - 4 x ^{2} - y ^{2}}{4}} y ( 2 - y ^{2} + x ^{2} )}{2}

= - 2 e^{1 - x^{2} - \frac{y ^{2}}{4}} y - \frac{e ^{\frac{4 - 4 x ^{2} - y ^{2}}{4}} y ( 2 - y ^{2} + x ^{2} )}{2}

\frac{d}{d x} (36 e^{- 2 x})

d er i v a t i v e f (x) = - 3 e^{4 x}

\int - 4 x^{- 5} d x

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

\int (2 x^{2} - x - 1) (3 x + 4) d x