derivative (4-t)(4+t^3)^{-1}

Lösung

ableitung von

(4 - t) (4 + t^{3})^{- 1}

\frac{2 t ^{3} - 12 t ^{2} - 4}{( t + 3 4 ) ^{2} ( t ^{2} - 3 4 t + 4 ^{\frac{2}{3}} ) ^{2}}

Schritte zur Lösung

\frac{d}{d t} ((4 - t) (4 + t^{3})^{- 1})

Wende die Produktregel an:

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

f = 4 - t, g = (4 + t^{3})^{- 1}

= \frac{d}{d t} (4 - t) (4 + t^{3})^{- 1} + \frac{d}{d t} ((4 + t^{3})^{- 1}) (4 - t)

\frac{d}{d t} (4 - t) = - 1

\frac{d}{d t} ((4 + t^{3})^{- 1}) = - \frac{3 t ^{2}}{( 4 + t ^{3} ) ^{2}}

= (- 1) (4 + t^{3})^{- 1} + (- \frac{3 t ^{2}}{( 4 + t ^{3} ) ^{2}}) (4 - t)

Vereinfache

(- 1) (4 + t^{3})^{- 1} + (- \frac{3 t ^{2}}{( 4 + t ^{3} ) ^{2}}) (4 - t) : \frac{2 t ^{3} - 12 t ^{2} - 4}{( t + 3 4 ) ^{2} ( t ^{2} - 3 4 t + 4 ^{\frac{2}{3}} ) ^{2}}

= \frac{2 t ^{3} - 12 t ^{2} - 4}{( t + 3 4 ) ^{2} ( t ^{2} - 3 4 t + 4 ^{\frac{2}{3}} ) ^{2}}

\frac{\partial}{\partial y} (\frac{4}{3} π (x^{3} - y^{3}))

x \to \infty lim (\frac{3}{5 x})

\int 9 e^{3 x} d x

\int x e^{3 x} + e^{x} d x

d er i v a t i v e f (x) = e^{2 x} (x^{3} + 1)