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

解

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

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

解答ステップ

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

y

を定数として扱う

連鎖法則を適用:

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

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

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

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

簡素化

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

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