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

解

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

\frac{6 x ^{2} + 4 x - 4}{x x}

解答ステップ

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

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

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

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

簡素化

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

= \frac{6 x ^{2} + 4 x - 4}{x x}