derivative of x^{2/3}(x+4^{1/2})

Lösung

\frac{d}{d x} (x^{\frac{2}{3}} (x + 4)^{\frac{1}{2}})

\frac{7 x + 16}{6 x ^{\frac{1}{3}} ( x + 4 ) ^{\frac{1}{2}}}

Schritte zur Lösung

\frac{d}{d x} (x^{\frac{2}{3}} (x + 4)^{\frac{1}{2}})

Wende die Produktregel an:

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

f = x^{\frac{2}{3}}, g = (x + 4)^{\frac{1}{2}}

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

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

Vereinfache

\frac{2}{3 x ^{\frac{1}{3}}} (x + 4)^{\frac{1}{2}} + \frac{1}{2 ( x + 4 ) ^{\frac{1}{2}}} x^{\frac{2}{3}} : \frac{7 x + 16}{6 x ^{\frac{1}{3}} ( x + 4 ) ^{\frac{1}{2}}}

= \frac{7 x + 16}{6 x ^{\frac{1}{3}} ( x + 4 ) ^{\frac{1}{2}}}

d er i v a t i v e f (x) = 8 x - x^{2}

d er i v a t i v e - x^{2} + 26 x + 920

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

n = 1 \sum \infty \frac{n}{n ^{3} + 5}

\frac{d}{d x} (x 4 - x)