derivative (t+2)^{2/3}(2t^2-3)^3

解答

导数

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

\frac{2 ( 2 t ^{2} - 3 ) ^{3} + 36 t ( 2 t ^{2} - 3 ) ^{2} ( t + 2 )}{3 ( t + 2 ) ^{\frac{1}{3}}}

求解步骤

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

使用微分乘法定则:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = (t + 2)^{\frac{2}{3}}, g = (2 t^{2} - 3)^{3}

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

\frac{d}{d t} ((t + 2)^{\frac{2}{3}}) = \frac{2}{3 ( t + 2 ) ^{\frac{1}{3}}}

\frac{d}{d t} ((2 t^{2} - 3)^{3}) = 12 t (2 t^{2} - 3)^{2}

= \frac{2}{3 ( t + 2 ) ^{\frac{1}{3}}} (2 t^{2} - 3)^{3} + 12 t (2 t^{2} - 3)^{2} (t + 2)^{\frac{2}{3}}

化简

\frac{2}{3 ( t + 2 ) ^{\frac{1}{3}}} (2 t^{2} - 3)^{3} + 12 t (2 t^{2} - 3)^{2} (t + 2)^{\frac{2}{3}} : \frac{2 ( 2 t ^{2} - 3 ) ^{3} + 36 t ( 2 t ^{2} - 3 ) ^{2} ( t + 2 )}{3 ( t + 2 ) ^{\frac{1}{3}}}

= \frac{2 ( 2 t ^{2} - 3 ) ^{3} + 36 t ( 2 t ^{2} - 3 ) ^{2} ( t + 2 )}{3 ( t + 2 ) ^{\frac{1}{3}}}

\frac{d}{d x} (f (x) \cdot \frac{1}{g ( x )})

\frac{y ^{^{'}} - e ^{- t} + 6}{y} = - 6

2 t y^{^{'}} - y = 2 t y^{3} cos (t)

t an g e n t y = 3 x^{4}, (- 2, 48)

\frac{d}{d x} (t^{x})