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

解

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

\frac{x ^{2} - 9}{2 x ^{2} x}

解答ステップ

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

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

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

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

簡素化

\frac{( 1 - \frac{3}{x ^{2}} ) x - \frac{1}{2 x} ( x + \frac{3}{x} )}{( x ) ^{2}} : \frac{x ^{2} - 9}{2 x ^{2} x}

= \frac{x ^{2} - 9}{2 x ^{2} x}