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

Lösung

ableitung von

h (t) = (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}}}

Schritte zur Lösung

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

Wende die Produktregel an:

(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}}

Vereinfache

\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 y}{d t} = (- csc (t) cot (t)) e^{- c s c (t)}

x \to \infty lim (- 3 x^{3})

x \to 0 - lim (1 - x^{2})

x \to 0 lim ((1 - 4 x)^{\frac{7}{x}})

n = 1 \sum \infty \frac{35}{- 6 ^{n}}