derivative (6x^2+3x+7)/(sqrt(x))

解

導関数

\frac{6 x ^{2} + 3 x + 7}{x}

\frac{18 x ^{2} + 3 x - 7}{2 x x}

解答ステップ

\frac{d}{d x} (\frac{6 x ^{2} + 3 x + 7}{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} ( 6 x ^{2} + 3 x + 7 ) x - \frac{d}{d x} ( x ) ( 6 x ^{2} + 3 x + 7 )}{( x ) ^{2}}

\frac{d}{d x} (6 x^{2} + 3 x + 7) = 12 x + 3

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

= \frac{( 12 x + 3 ) x - \frac{1}{2 x} ( 6 x ^{2} + 3 x + 7 )}{( x ) ^{2}}

簡素化

\frac{( 12 x + 3 ) x - \frac{1}{2 x} ( 6 x ^{2} + 3 x + 7 )}{( x ) ^{2}} : \frac{18 x ^{2} + 3 x - 7}{2 x x}

= \frac{18 x ^{2} + 3 x - 7}{2 x x}