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

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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