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

解

\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}}}

解答ステップ

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

y, z

を定数として扱う

指数の規則を適用する:

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})})

連鎖法則を適用:

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}})

簡素化

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}}}