derivative of (1+sqrt(3x)/(1-sqrt(3x)))

解

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

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

解答ステップ

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

\frac{d}{d x} (1 + 3 x) = \frac{3}{2 x}

\frac{d}{d x} (1 - 3 x) = - \frac{3}{2 x}

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

簡素化

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

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