derivative ((x^2-x-2))/((x^2-6x+9))

解答

导数

\frac{( x ^{2} - x - 2 )}{( x ^{2} - 6 x + 9 )}

- \frac{5 x - 7}{( x - 3 ) ^{3}}

求解步骤

\frac{d}{d x} (\frac{x ^{2} - x - 2}{x ^{2} - 6 x + 9})

使用微分除法定则:

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

= \frac{\frac{d}{d x} ( x ^{2} - x - 2 ) ( x ^{2} - 6 x + 9 ) - \frac{d}{d x} ( x ^{2} - 6 x + 9 ) ( x ^{2} - x - 2 )}{( x ^{2} - 6 x + 9 ) ^{2}}

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

\frac{d}{d x} (x^{2} - 6 x + 9) = 2 x - 6

= \frac{( 2 x - 1 ) ( x ^{2} - 6 x + 9 ) - ( 2 x - 6 ) ( x ^{2} - x - 2 )}{( x ^{2} - 6 x + 9 ) ^{2}}

化简

\frac{( 2 x - 1 ) ( x ^{2} - 6 x + 9 ) - ( 2 x - 6 ) ( x ^{2} - x - 2 )}{( x ^{2} - 6 x + 9 ) ^{2}} : - \frac{5 x - 7}{( x - 3 ) ^{3}}

= - \frac{5 x - 7}{( x - 3 ) ^{3}}