derivative f(x)=(2-ln(x))/(2xsqrt(x))

解

導関数

f (x) = \frac{2 - ln ( x )}{2 x x}

\frac{- 8 + 3 ln ( x )}{4 x ^{\frac{5}{2}}}

解答ステップ

\frac{d}{d x} (\frac{2 - ln ( x )}{2 x x})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

商の法則を適用:

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

= \frac{1}{2} \cdot \frac{\frac{d}{d x} ( 2 - ln ( x ) ) x x - \frac{d}{d x} ( x x ) ( 2 - ln ( x ) )}{( x x ) ^{2}}

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

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

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

簡素化

\frac{1}{2} \cdot \frac{( - \frac{1}{x} ) x x - \frac{3 x}{2} ( 2 - ln ( x ) )}{( x x ) ^{2}} : \frac{- 8 + 3 ln ( x )}{4 x ^{\frac{5}{2}}}

= \frac{- 8 + 3 ln ( x )}{4 x ^{\frac{5}{2}}}