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

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

\frac{d x}{d x} = 1

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

Vereinfache

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

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

\int \frac{t ^{3}}{4 3 + t ^{4}} d t

\int_{0}^{1} e^{- s t} (t) d t

49 y^{^{''}} - 70 y^{^{'}} + 25 y = 0, y (0) = 2, y^{^{'}} (0) = 3

k = 1 \sum \infty (sin (100))^{k}

\int ln (x) \cdot x d x