(\partial)/(\partial x)((ax+by)/((cx+dy)))

解

\frac{\partial}{\partial x} (\frac{a x + b y}{( c x + d y )})

\frac{- b yc + a y d}{( x c + y d ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{a x + b y}{( c x + d y )})

a, b, y, c, d

を定数として扱う

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{\partial}{\partial x} ( a x + b y ) ( c x + d y ) - \frac{\partial}{\partial x} ( c x + d y ) ( a x + b y )}{( c x + d y ) ^{2}}

\frac{\partial}{\partial x} (a x + b y) = a

\frac{\partial}{\partial x} (c x + d y) = c

= \frac{a ( c x + d y ) - c ( a x + b y )}{( c x + d y ) ^{2}}

拡張

a (c x + d y) - c (a x + b y) : - b yc + a y d

= \frac{- b yc + a y d}{( x c + y d ) ^{2}}