derivative h(t)=(t+5)^{2/3}(3t^2-5)^3

解答

导数

h (t) = (t + 5)^{\frac{2}{3}} (3 t^{2} - 5)^{3}

\frac{2 ( 3 t ^{2} - 5 ) ^{3} + 54 t ( 3 t ^{2} - 5 ) ^{2} ( t + 5 )}{3 ( t + 5 ) ^{\frac{1}{3}}}

求解步骤

\frac{d}{d t} ((t + 5)^{\frac{2}{3}} (3 t^{2} - 5)^{3})

使用微分乘法定则:

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

f = (t + 5)^{\frac{2}{3}}, g = (3 t^{2} - 5)^{3}

= \frac{d}{d t} ((t + 5)^{\frac{2}{3}}) (3 t^{2} - 5)^{3} + \frac{d}{d t} ((3 t^{2} - 5)^{3}) (t + 5)^{\frac{2}{3}}

\frac{d}{d t} ((t + 5)^{\frac{2}{3}}) = \frac{2}{3 ( t + 5 ) ^{\frac{1}{3}}}

\frac{d}{d t} ((3 t^{2} - 5)^{3}) = 18 t (3 t^{2} - 5)^{2}

= \frac{2}{3 ( t + 5 ) ^{\frac{1}{3}}} (3 t^{2} - 5)^{3} + 18 t (3 t^{2} - 5)^{2} (t + 5)^{\frac{2}{3}}

化简

\frac{2}{3 ( t + 5 ) ^{\frac{1}{3}}} (3 t^{2} - 5)^{3} + 18 t (3 t^{2} - 5)^{2} (t + 5)^{\frac{2}{3}} : \frac{2 ( 3 t ^{2} - 5 ) ^{3} + 54 t ( 3 t ^{2} - 5 ) ^{2} ( t + 5 )}{3 ( t + 5 ) ^{\frac{1}{3}}}

= \frac{2 ( 3 t ^{2} - 5 ) ^{3} + 54 t ( 3 t ^{2} - 5 ) ^{2} ( t + 5 )}{3 ( t + 5 ) ^{\frac{1}{3}}}

x \to 9 lim (\frac{5}{( x - 9 )})

7 y y^{^{'}} = x

x \to (a + 1) lim (x^{2} - 1)

\frac{d}{d x} (\frac{3}{2} x)

l a pl a ce t r an s f or m cos (3 t - 1)