derivative of \sqrt[5]{(x^3-2/(x^3+2)})

Lösung

\frac{12 x ^{2}}{5 ( x ^{3} - 2 ) ^{\frac{4}{5}} ( x ^{3} + 2 ) ^{\frac{6}{5}}}

Schritte zur Lösung

Wende die Kettenregel an:

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

= \frac{1}{5 ( \frac{x ^{3} - 2}{x ^{3} + 2} ) ^{\frac{4}{5}}} \frac{d}{d x} (\frac{x ^{3} - 2}{x ^{3} + 2})

\frac{d}{d x} (\frac{x ^{3} - 2}{x ^{3} + 2}) = \frac{12 x ^{2}}{( x ^{3} + 2 ) ^{2}}

= \frac{1}{5 ( \frac{x ^{3} - 2}{x ^{3} + 2} ) ^{\frac{4}{5}}} \cdot \frac{12 x ^{2}}{( x ^{3} + 2 ) ^{2}}

Vereinfache

\frac{1}{5 ( \frac{x ^{3} - 2}{x ^{3} + 2} ) ^{\frac{4}{5}}} \cdot \frac{12 x ^{2}}{( x ^{3} + 2 ) ^{2}} : \frac{12 x ^{2}}{5 ( x ^{3} - 2 ) ^{\frac{4}{5}} ( x ^{3} + 2 ) ^{\frac{6}{5}}}

= \frac{12 x ^{2}}{5 ( x ^{3} - 2 ) ^{\frac{4}{5}} ( x ^{3} + 2 ) ^{\frac{6}{5}}}