derivative of \sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{x}}}

Lösung

\frac{d}{d x} (3 1 + 3 1 + 3 x)

\frac{1}{27 x ^{\frac{2}{3}} ( 1 + 3 1 + 3 x ) ^{\frac{2}{3}} ( 3 x + 1 ) ^{\frac{2}{3}}}

Schritte zur Lösung

\frac{d}{d x} (3 1 + 3 1 + 3 x)

Wende die Kettenregel an:

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

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

\frac{d}{d x} (1 + 3 1 + 3 x) = \frac{1}{9 x ^{\frac{2}{3}} ( 3 x + 1 ) ^{\frac{2}{3}}}

= \frac{1}{3 ( 1 + 3 1 + 3 x ) ^{\frac{2}{3}}} \cdot \frac{1}{9 x ^{\frac{2}{3}} ( 3 x + 1 ) ^{\frac{2}{3}}}

Vereinfache

\frac{1}{3 ( 1 + 3 1 + 3 x ) ^{\frac{2}{3}}} \cdot \frac{1}{9 x ^{\frac{2}{3}} ( 3 x + 1 ) ^{\frac{2}{3}}} : \frac{1}{27 x ^{\frac{2}{3}} ( 1 + 3 1 + 3 x ) ^{\frac{2}{3}} ( 3 x + 1 ) ^{\frac{2}{3}}}

= \frac{1}{27 x ^{\frac{2}{3}} ( 1 + 3 1 + 3 x ) ^{\frac{2}{3}} ( 3 x + 1 ) ^{\frac{2}{3}}}

\int_{- 2}^{2} \frac{1}{x ^{2} - 1} d x

\int \frac{18}{x} d x

t an g e n t f (x) = x^{2} - 2 x, at x = 3

\frac{\partial}{\partial y} (e^{x} + y e^{x y})

\frac{\partial}{\partial x} (e^{y} + x + x^{2} + 2)