derivative (5ln(4x))/(2-3x)

解

導関数

\frac{5 ln ( 4 x )}{2 - 3 x}

\frac{5 ( 2 - 3 x + 3 x ln ( 4 x ) )}{x ( 2 - 3 x ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{5 ln ( 4 x )}{2 - 3 x})

定数を除去:

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

= 5 \frac{d}{d x} (\frac{ln ( 4 x )}{2 - 3 x})

商の法則を適用:

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

= 5 \cdot \frac{\frac{d}{d x} ( ln ( 4 x ) ) ( 2 - 3 x ) - \frac{d}{d x} ( 2 - 3 x ) ln ( 4 x )}{( 2 - 3 x ) ^{2}}

\frac{d}{d x} (ln (4 x)) = \frac{1}{x}

\frac{d}{d x} (2 - 3 x) = - 3

= 5 \cdot \frac{\frac{1}{x} ( 2 - 3 x ) - ( - 3 ) ln ( 4 x )}{( 2 - 3 x ) ^{2}}

簡素化

5 \cdot \frac{\frac{1}{x} ( 2 - 3 x ) - ( - 3 ) ln ( 4 x )}{( 2 - 3 x ) ^{2}} : \frac{5 ( 2 - 3 x + 3 x ln ( 4 x ) )}{x ( 2 - 3 x ) ^{2}}

= \frac{5 ( 2 - 3 x + 3 x ln ( 4 x ) )}{x ( 2 - 3 x ) ^{2}}