derivative h(t)=(t+5)^{2/3}(3t^2-1)^3

Lösung

ableitung von

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

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

Schritte zur Lösung

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

Wende die Produktregel an:

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

f = (t + 5)^{\frac{2}{3}}, g = (3 t^{2} - 1)^{3}

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

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

Vereinfache

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

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

t an g e n t y = sin (4 x)

\int (cos (x))^{3} sin (x) d x

\int_{0}^{3} ((30 - y) - y^{\frac{1}{3}}) d y

\frac{d}{d x} (sin (x) - e^{x})

y^{^{''}} + 2 y^{^{'}} + 2 y = sin (2 t), y (0) = 0, y^{^{'}} (0) = 0