derivative (t^3+3t)/(t^2-4t+3)

解答

导数

\frac{t ^{3} + 3 t}{t ^{2} - 4 t + 3}

\frac{t ^{4} - 8 t ^{3} + 6 t ^{2} + 9}{( t ^{2} - 4 t + 3 ) ^{2}}

求解步骤

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

使用微分除法定则:

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

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

\frac{d}{d t} (t^{3} + 3 t) = 3 t^{2} + 3

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

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

乘开

(3 t^{2} + 3) (t^{2} - 4 t + 3) - (2 t - 4) (t^{3} + 3 t) : t^{4} - 8 t^{3} + 6 t^{2} + 9

= \frac{t ^{4} - 8 t ^{3} + 6 t ^{2} + 9}{( t ^{2} - 4 t + 3 ) ^{2}}

\int \frac{7}{x ( x - 7 ) ^{2}} d x

\frac{d}{d x} ((x - 100)^{3} + (x - 100)^{2} + 500)

\int 1 - y^{2} d y

\frac{d}{d x} ((x^{3} + 7)^{\frac{2}{3}})

d er i v a t i v e g (x) = 8 - x