(\partial)/(\partial y)(ln((y/x)^2))

解

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

\frac{2}{y}

解答ステップ

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

簡素化

ln ((\frac{y}{x})^{2}) : ln (\frac{y ^{2}}{x ^{2}})

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

x

を定数として扱う

連鎖法則を適用:

\frac{1}{\frac{y ^{2}}{x ^{2}}} \frac{\partial}{\partial y} (\frac{y ^{2}}{x ^{2}})

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

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

= \frac{1}{\frac{y ^{2}}{x ^{2}}} \cdot \frac{2 y}{x ^{2}}

簡素化

\frac{1}{\frac{y ^{2}}{x ^{2}}} \cdot \frac{2 y}{x ^{2}} : \frac{2}{y}

= \frac{2}{y}