(\partial)/(\partial y)(e^{xyz^5}yz^5)

Lösung

\frac{\partial}{\partial y} (e^{x y z^{5}} y z^{5})

z^{5} (e^{x y z^{5}} x y z^{5} + e^{x y z^{5}})

Schritte zur Lösung

\frac{\partial}{\partial y} (e^{x y z^{5}} y z^{5})

Behandle

x, z

als Konstanten

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= z^{5} \frac{\partial}{\partial y} (e^{x y z^{5}} y)

Wende die Produktregel an:

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

f = e^{x y z^{5}}, g = y

= z^{5} (\frac{\partial}{\partial y} (e^{x y z^{5}}) y + \frac{\partial}{\partial y} (y) e^{x y z^{5}})

\frac{\partial}{\partial y} (e^{x y z^{5}}) = e^{x y z^{5}} x z^{5}

\frac{\partial}{\partial y} (y) = 1

= z^{5} (e^{x y z^{5}} x z^{5} y + 1 \cdot e^{x y z^{5}})

Vereinfache

= z^{5} (e^{x y z^{5}} x y z^{5} + e^{x y z^{5}})