derivative f(x)=(8x^2+6x+4)/(sqrt(x))

解

導関数

f (x) = \frac{8 x ^{2} + 6 x + 4}{x}

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

解答ステップ

\frac{d}{d x} (\frac{8 x ^{2} + 6 x + 4}{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} ( 8 x ^{2} + 6 x + 4 ) x - \frac{d}{d x} ( x ) ( 8 x ^{2} + 6 x + 4 )}{( x ) ^{2}}

\frac{d}{d x} (8 x^{2} + 6 x + 4) = 16 x + 6

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

= \frac{( 16 x + 6 ) x - \frac{1}{2 x} ( 8 x ^{2} + 6 x + 4 )}{( x ) ^{2}}

簡素化

\frac{( 16 x + 6 ) x - \frac{1}{2 x} ( 8 x ^{2} + 6 x + 4 )}{( x ) ^{2}} : \frac{12 x ^{2} + 3 x - 2}{x x}

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