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

解

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

\frac{3 x ^{2} + 2 x - 4}{x x}

解答ステップ

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

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

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

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

簡素化

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

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