derivative (x^2-6x+7)/(x^2+9)

解答

导数

\frac{x ^{2} - 6 x + 7}{x ^{2} + 9}

\frac{6 x ^{2} + 4 x - 54}{( x ^{2} + 9 ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{x ^{2} - 6 x + 7}{x ^{2} + 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} - 6 x + 7 ) ( x ^{2} + 9 ) - \frac{d}{d x} ( x ^{2} + 9 ) ( x ^{2} - 6 x + 7 )}{( x ^{2} + 9 ) ^{2}}

\frac{d}{d x} (x^{2} - 6 x + 7) = 2 x - 6

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

= \frac{( 2 x - 6 ) ( x ^{2} + 9 ) - 2 x ( x ^{2} - 6 x + 7 )}{( x ^{2} + 9 ) ^{2}}

乘开

(2 x - 6) (x^{2} + 9) - 2 x (x^{2} - 6 x + 7) : 6 x^{2} + 4 x - 54

= \frac{6 x ^{2} + 4 x - 54}{( x ^{2} + 9 ) ^{2}}

\int_{1}^{3} \frac{3}{x} - \frac{3}{x ^{2}} d x

\int x^{3} e^{- 0.25 x} d x

\frac{d}{d x} (3 \cdot 3 x)

f^{^{'}} (x) = 3 x

y^{^{'}} + y = \frac{e ^{x}}{y ^{2}}