(\partial)/(\partial z)(3x^{y/z})

Lösung

\frac{\partial}{\partial z} (3 x^{\frac{y}{z}})

- \frac{3 y x ^{\frac{y}{z}} ln ( x )}{z ^{2}}

Schritte zur Lösung

\frac{\partial}{\partial z} (3 x^{\frac{y}{z}})

Behandle

x, y

als Konstanten

Entferne die Konstante:

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

= 3 \frac{\partial}{\partial z} (x^{\frac{y}{z}})

Wende Exponentenregel an:

a^{b} = e^{b l n (a)}

x^{\frac{y}{z}} = e^{\frac{y}{z} l n (x)}

= 3 \frac{\partial}{\partial z} (e^{\frac{y}{z} l n (x)})

Wende die Kettenregel an:

e^{\frac{y}{z} l n (x)} \frac{\partial}{\partial z} (\frac{y}{z} ln (x))

= e^{\frac{y}{z} l n (x)} \frac{\partial}{\partial z} (\frac{y}{z} ln (x))

\frac{\partial}{\partial z} (\frac{y}{z} ln (x)) = - \frac{y ln ( x )}{z ^{2}}

= 3 e^{\frac{y}{z} l n (x)} (- \frac{y ln ( x )}{z ^{2}})

Vereinfache

3 e^{\frac{y}{z} l n (x)} (- \frac{y ln ( x )}{z ^{2}}) : - \frac{3 y x ^{\frac{y}{z}} ln ( x )}{z ^{2}}

= - \frac{3 y x ^{\frac{y}{z}} ln ( x )}{z ^{2}}

d er i v a t i v e e^{x} \frac{( 2 x + 5 ) ^{2}}{x - 2}

y^{^{'}} = - 2 y + x

d er i v a t i v e 3 sin^{3} (x)

t a y l or ln (- 3 x) + 2

\frac{\partial}{\partial x} (e^{0})