derivative of x^{2/3}(x-2)

Lösung

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

\frac{5 x - 4}{3 x ^{\frac{1}{3}}}

Schritte zur Lösung

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

Wende die Produktregel an:

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

f = x^{\frac{2}{3}}, g = x - 2

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

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

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

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

Vereinfache

\frac{2}{3 x ^{\frac{1}{3}}} (x - 2) + 1 \cdot x^{\frac{2}{3}} : \frac{5 x - 4}{3 x ^{\frac{1}{3}}}

= \frac{5 x - 4}{3 x ^{\frac{1}{3}}}

d er i v a t i v e ln (sec (2 x) + tan (2 x))

d er i v a t i v e f (x) = 3 x^{2} e^{x} + e^{x} (x^{3} + 9)

y \frac{d y}{d x} = \frac{4}{5} x

t an g e n t f (x) = 4 sec (x), at x = \frac{7 π}{6}

\int \frac{2 x ^{2}}{( x - 1 ) ^{2} + 4} d x