derivative of x(8-x^{1/3})

Lösung

\frac{d}{d x} (x (8 - x)^{\frac{1}{3}})

\frac{- 4 x + 24}{3 ( 8 - x ) ^{\frac{2}{3}}}

Schritte zur Lösung

\frac{d}{d x} (x (8 - x)^{\frac{1}{3}})

Wende die Produktregel an:

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

f = x, g = (8 - x)^{\frac{1}{3}}

= \frac{d x}{d x} (8 - x)^{\frac{1}{3}} + \frac{d}{d x} ((8 - x)^{\frac{1}{3}}) x

\frac{d x}{d x} = 1

\frac{d}{d x} ((8 - x)^{\frac{1}{3}}) = - \frac{1}{3 ( 8 - x ) ^{\frac{2}{3}}}

= 1 \cdot (8 - x)^{\frac{1}{3}} + (- \frac{1}{3 ( 8 - x ) ^{\frac{2}{3}}}) x

Vereinfache

1 \cdot (8 - x)^{\frac{1}{3}} + (- \frac{1}{3 ( 8 - x ) ^{\frac{2}{3}}}) x : \frac{- 4 x + 24}{3 ( 8 - x ) ^{\frac{2}{3}}}

= \frac{- 4 x + 24}{3 ( 8 - x ) ^{\frac{2}{3}}}

d er i v a t i v e 5 2 x^{3} + 8 x

\int - 3 e^{- 3 t} d t

\frac{\partial}{\partial x} (∣ x ∣)

\int_{- 8}^{10} (x + 10) d x

\frac{d}{d x} (2 π x)