derivative of x/(3x+4ln(x))

解

\frac{d}{d x} (\frac{x}{3 x + 4 ln ( x )})

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

解答ステップ

\frac{d}{d x} (\frac{x}{3 x + 4 ln ( x )})

商の法則を適用:

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

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

\frac{d x}{d x} = 1

\frac{d}{d x} (3 x + 4 ln (x)) = 3 + \frac{4}{x}

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

簡素化

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

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