(\partial)/(\partial x)((y/z)^{1/x})

Lösung

\frac{\partial}{\partial x} ((\frac{y}{z})^{\frac{1}{x}})

- \frac{y ^{\frac{1}{x}} ln ( \frac{y}{z} )}{x ^{2} z ^{\frac{1}{x}}}

Schritte zur Lösung

\frac{\partial}{\partial x} ((\frac{y}{z})^{\frac{1}{x}})

Behandle

y, z

als Konstanten

Wende Exponentenregel an:

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

(\frac{y}{z})^{\frac{1}{x}} = e^{\frac{1}{x} l n (\frac{y}{z})}

= \frac{\partial}{\partial x} (e^{\frac{1}{x} l n (\frac{y}{z})})

Wende die Kettenregel an:

e^{\frac{1}{x} l n (\frac{y}{z})} \frac{\partial}{\partial x} (\frac{1}{x} ln (\frac{y}{z}))

= e^{\frac{1}{x} l n (\frac{y}{z})} \frac{\partial}{\partial x} (\frac{1}{x} ln (\frac{y}{z}))

\frac{\partial}{\partial x} (\frac{1}{x} ln (\frac{y}{z})) = - \frac{ln ( \frac{y}{z} )}{x ^{2}}

= e^{\frac{1}{x} l n (\frac{y}{z})} (- \frac{ln ( \frac{y}{z} )}{x ^{2}})

Vereinfache

e^{\frac{1}{x} l n (\frac{y}{z})} (- \frac{ln ( \frac{y}{z} )}{x ^{2}}) : - \frac{y ^{\frac{1}{x}} ln ( \frac{y}{z} )}{x ^{2} z ^{\frac{1}{x}}}

= - \frac{y ^{\frac{1}{x}} ln ( \frac{y}{z} )}{x ^{2} z ^{\frac{1}{x}}}

y^{^{''}} + 3 y = 48 x^{2} e^{3 x}

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

\frac{\partial}{\partial x} (x + y + x^{3} + y^{2})

d er i v a t i v e x^{3} sin (7 x)

\frac{d}{d x} (6 + 7 x)