derivative of (ax+by(x)/(cx+dy))

解

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

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

解答ステップ

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

商の法則を適用:

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

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

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

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

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

拡張

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

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