derivative f(x)=(3x+1)/(5+x)

解

導関数

f (x) = \frac{3 x + 1}{5 + x}

\frac{14}{( 5 + x ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{3 x + 1}{5 + 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} ( 3 x + 1 ) ( 5 + x ) - \frac{d}{d x} ( 5 + x ) ( 3 x + 1 )}{( 5 + x ) ^{2}}

\frac{d}{d x} (3 x + 1) = 3

\frac{d}{d x} (5 + x) = 1

= \frac{3 ( 5 + x ) - 1 \cdot ( 3 x + 1 )}{( 5 + x ) ^{2}}

簡素化

\frac{3 ( 5 + x ) - 1 \cdot ( 3 x + 1 )}{( 5 + x ) ^{2}} : \frac{14}{( 5 + x ) ^{2}}

= \frac{14}{( 5 + x ) ^{2}}