d/(d{y)}({x}^{({z})/({y)}})

解

\frac{d}{d y} (x^{\frac{z}{y}})

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

解答ステップ

\frac{d}{d y} (x^{\frac{z}{y}})

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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