derivative of (x-3/(x^2+4x+4))

解

\frac{d}{d x} (\frac{x - 3}{x ^{2} + 4 x + 4})

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

解答ステップ

\frac{d}{d x} (\frac{x - 3}{x ^{2} + 4 x + 4})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

\frac{d}{d x} (x - 3) = 1

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

= \frac{1 \cdot ( x ^{2} + 4 x + 4 ) - ( 2 x + 4 ) ( x - 3 )}{( x ^{2} + 4 x + 4 ) ^{2}}

簡素化

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

= - \frac{x - 8}{( x + 2 ) ^{3}}