derivative of (sqrt(x)/(6+x))

解

\frac{d}{d x} (\frac{x}{6 + x})

\frac{6 - x}{2 x ( 6 + x ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x}{6 + 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} ( x ) ( 6 + x ) - \frac{d}{d x} ( 6 + x ) x}{( 6 + x ) ^{2}}

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

\frac{d}{d x} (6 + x) = 1

= \frac{\frac{1}{2 x} ( 6 + x ) - 1 \cdot x}{( 6 + x ) ^{2}}

簡素化

\frac{\frac{1}{2 x} ( 6 + x ) - 1 \cdot x}{( 6 + x ) ^{2}} : \frac{6 - x}{2 x ( 6 + x ) ^{2}}

= \frac{6 - x}{2 x ( 6 + x ) ^{2}}