derivative f(x)=(4x^2-x+3)/(sqrt(x))

解

導関数

f (x) = \frac{4 x ^{2} - x + 3}{x}

\frac{12 x ^{2} - x - 3}{2 x x}

解答ステップ

\frac{d}{d x} (\frac{4 x ^{2} - x + 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} ( 4 x ^{2} - x + 3 ) x - \frac{d}{d x} ( x ) ( 4 x ^{2} - x + 3 )}{( x ) ^{2}}

\frac{d}{d x} (4 x^{2} - x + 3) = 8 x - 1

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

= \frac{( 8 x - 1 ) x - \frac{1}{2 x} ( 4 x ^{2} - x + 3 )}{( x ) ^{2}}

簡素化

\frac{( 8 x - 1 ) x - \frac{1}{2 x} ( 4 x ^{2} - x + 3 )}{( x ) ^{2}} : \frac{12 x ^{2} - x - 3}{2 x x}

= \frac{12 x ^{2} - x - 3}{2 x x}