derivative of x^{2/3}y^{1/3}

Lösung

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

\frac{2 y + x \frac{d y}{d x}}{3 x ^{\frac{1}{3}} y ^{\frac{2}{3}}}

Schritte zur Lösung

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

Wende die Produktregel an:

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

f = x^{\frac{2}{3}}, g = y^{\frac{1}{3}}

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

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

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

= \frac{2}{3 x ^{\frac{1}{3}}} y^{\frac{1}{3}} + \frac{\frac{d y}{d x}}{3 y ^{\frac{2}{3}}} x^{\frac{2}{3}}

Vereinfache

\frac{2}{3 x ^{\frac{1}{3}}} y^{\frac{1}{3}} + \frac{\frac{d y}{d x}}{3 y ^{\frac{2}{3}}} x^{\frac{2}{3}} : \frac{2 y + x \frac{d y}{d x}}{3 x ^{\frac{1}{3}} y ^{\frac{2}{3}}}

= \frac{2 y + x \frac{d y}{d x}}{3 x ^{\frac{1}{3}} y ^{\frac{2}{3}}}

\int (6 x - 4) d x

f (x) = arctan (\frac{1}{x})

f (x) = 3 x - 5

\int \frac{- x}{y} d x

d er i v a t i v e f (x) = 9 - 4 x