(\partial)/(\partial x)(ln(x/y+y/x))

解

\frac{\partial}{\partial x} (ln (\frac{x}{y} + \frac{y}{x}))

\frac{x ^{2} - y ^{2}}{x ( x ^{2} + y ^{2} )}

解答ステップ

\frac{\partial}{\partial x} (ln (\frac{x}{y} + \frac{y}{x}))

y

を定数として扱う

連鎖法則を適用:

\frac{1}{\frac{x}{y} + \frac{y}{x}} \frac{\partial}{\partial x} (\frac{x}{y} + \frac{y}{x})

= \frac{1}{\frac{x}{y} + \frac{y}{x}} \frac{\partial}{\partial x} (\frac{x}{y} + \frac{y}{x})

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

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

簡素化

\frac{1}{\frac{x}{y} + \frac{y}{x}} (\frac{1}{y} - \frac{y}{x ^{2}}) : \frac{x ^{2} - y ^{2}}{x ( x ^{2} + y ^{2} )}

= \frac{x ^{2} - y ^{2}}{x ( x ^{2} + y ^{2} )}