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

解

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

\frac{3 ( x + 1 )}{x ( x + 6 ) x + 6}

解答ステップ

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

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

\frac{d}{d x} (x - 6) = \frac{1}{2 x}

\frac{d}{d x} (x + 6) = \frac{1}{2 x + 6}

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

簡素化

\frac{\frac{1}{2 x} x + 6 - \frac{1}{2 x + 6} ( x - 6 )}{( x + 6 ) ^{2}} : \frac{3 ( x + 1 )}{x ( x + 6 ) x + 6}

= \frac{3 ( x + 1 )}{x ( x + 6 ) x + 6}