d/(dy)(1/y e^{xy})

Lösung

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

- \frac{e ^{y x}}{y ^{2}} + \frac{e ^{y x} x}{y}

Schritte zur Lösung

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

Behandle

x

als Konstante

Wende die Produktregel an:

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

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

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

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

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

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

Vereinfache

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

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

y^{2} = (x y y^{^{'}} + 1) ln (x)

y^{^{'}} = 1 - y

t^{2} y^{^{'}} + 6 t y - y^{3} = 0

y^{^{''}} - 2 y^{^{'}} + 4 y = 0

d er i v a t i v e e^{x^{2} - 4 x}