derivative of ((x+3(x+1))/(x^2-49))

解答

\frac{d}{d x} (\frac{( x + 3 ) ( x + 1 )}{x ^{2} - 49})

\frac{- 4 x ^{2} - 104 x - 196}{( x ^{2} - 49 ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{( x + 3 ) ( x + 1 )}{x ^{2} - 49})

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 + 3 ) ( x + 1 ) ) ( x ^{2} - 49 ) - \frac{d}{d x} ( x ^{2} - 49 ) ( x + 3 ) ( x + 1 )}{( x ^{2} - 49 ) ^{2}}

\frac{d}{d x} ((x + 3) (x + 1)) = 2 x + 4

\frac{d}{d x} (x^{2} - 49) = 2 x

= \frac{( 2 x + 4 ) ( x ^{2} - 49 ) - 2 x ( x + 3 ) ( x + 1 )}{( x ^{2} - 49 ) ^{2}}

乘开

(2 x + 4) (x^{2} - 49) - 2 x (x + 3) (x + 1) : - 4 x^{2} - 104 x - 196

= \frac{- 4 x ^{2} - 104 x - 196}{( x ^{2} - 49 ) ^{2}}