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

Lösung

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

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

Schritte zur Lösung

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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