derivative (9e^{9x})/(8x+3)

解

導関数

\frac{9 e ^{9 x}}{8 x + 3}

\frac{9 ( 72 e ^{9 x} x + 19 e ^{9 x} )}{( 8 x + 3 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{9 e ^{9 x}}{8 x + 3})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 9 \frac{d}{d x} (\frac{e ^{9 x}}{8 x + 3})

商の法則を適用:

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

= 9 \cdot \frac{\frac{d}{d x} ( e ^{9 x} ) ( 8 x + 3 ) - \frac{d}{d x} ( 8 x + 3 ) e ^{9 x}}{( 8 x + 3 ) ^{2}}

\frac{d}{d x} (e^{9 x}) = e^{9 x} \cdot 9

\frac{d}{d x} (8 x + 3) = 8

= 9 \cdot \frac{e ^{9 x} \cdot 9 ( 8 x + 3 ) - 8 e ^{9 x}}{( 8 x + 3 ) ^{2}}

簡素化

9 \cdot \frac{e ^{9 x} \cdot 9 ( 8 x + 3 ) - 8 e ^{9 x}}{( 8 x + 3 ) ^{2}} : \frac{9 ( 72 e ^{9 x} x + 19 e ^{9 x} )}{( 8 x + 3 ) ^{2}}

= \frac{9 ( 72 e ^{9 x} x + 19 e ^{9 x} )}{( 8 x + 3 ) ^{2}}