derivative (\sqrt[5]{x^2}+\sqrt[4]{x^5})/(2x)

Lösung

ableitung von

\frac{5 x ^{2} + 4 x ^{5}}{2 x}

\frac{8 x ^{2} ( x ^{5} ) ^{\frac{3}{4}} + 25 x ^{5} ( x ^{2} ) ^{\frac{4}{5}} - 20 5 x ^{2} ( x ^{2} ) ^{\frac{4}{5}} ( x ^{5} ) ^{\frac{3}{4}} - 20 4 x 4 x ^{4} ( x ^{2} ) ^{\frac{4}{5}} ( x ^{5} ) ^{\frac{3}{4}}}{40 x ^{2} ( x ^{2} ) ^{\frac{4}{5}} ( x ^{5} ) ^{\frac{3}{4}}}

Schritte zur Lösung

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

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

\frac{1}{2} \cdot \frac{( \frac{2 x}{5 ( x ^{2} ) ^{\frac{4}{5}}} + \frac{5 x ^{4}}{4 ( x ^{5} ) ^{\frac{3}{4}}} ) x - 1 \cdot ( 5 x ^{2} + 4 x ^{5} )}{x ^{2}} : \frac{8 x ^{2} ( x ^{5} ) ^{\frac{3}{4}} + 25 x ^{5} ( x ^{2} ) ^{\frac{4}{5}} - 20 5 x ^{2} ( x ^{2} ) ^{\frac{4}{5}} ( x ^{5} ) ^{\frac{3}{4}} - 20 4 x 4 x ^{4} ( x ^{2} ) ^{\frac{4}{5}} ( x ^{5} ) ^{\frac{3}{4}}}{40 x ^{2} ( x ^{2} ) ^{\frac{4}{5}} ( x ^{5} ) ^{\frac{3}{4}}}

= \frac{8 x ^{2} ( x ^{5} ) ^{\frac{3}{4}} + 25 x ^{5} ( x ^{2} ) ^{\frac{4}{5}} - 20 5 x ^{2} ( x ^{2} ) ^{\frac{4}{5}} ( x ^{5} ) ^{\frac{3}{4}} - 20 4 x 4 x ^{4} ( x ^{2} ) ^{\frac{4}{5}} ( x ^{5} ) ^{\frac{3}{4}}}{40 x ^{2} ( x ^{2} ) ^{\frac{4}{5}} ( x ^{5} ) ^{\frac{3}{4}}}

\frac{\partial ^{2}}{\partial y \partial x} (x y e^{- 6 y})

y^{^{'}} - y = 7

\frac{\partial}{\partial u} (sin (u))

d er i v a t i v e f (x) = \frac{c ^{5} - x ^{5}}{c ^{5} + c ^{5}}, c

d er i v a t i v e 3 + 5 sin (s)