derivative g(x)=(9ln(x))/(4x+5)

解

導関数

g (x) = \frac{9 ln ( x )}{4 x + 5}

\frac{9 ( 4 x + 5 - 4 x ln ( x ) )}{x ( 4 x + 5 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{9 ln ( x )}{4 x + 5})

定数を除去:

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

= 9 \frac{d}{d x} (\frac{ln ( x )}{4 x + 5})

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

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

\frac{d}{d x} (4 x + 5) = 4

= 9 \cdot \frac{\frac{1}{x} ( 4 x + 5 ) - 4 ln ( x )}{( 4 x + 5 ) ^{2}}

簡素化

9 \cdot \frac{\frac{1}{x} ( 4 x + 5 ) - 4 ln ( x )}{( 4 x + 5 ) ^{2}} : \frac{9 ( 4 x + 5 - 4 x ln ( x ) )}{x ( 4 x + 5 ) ^{2}}

= \frac{9 ( 4 x + 5 - 4 x ln ( x ) )}{x ( 4 x + 5 ) ^{2}}