derivative (1-x^2+x^4)/(x^2*sqrt(x))

Lösung

ableitung von

\frac{1 - x ^{2} + x ^{4}}{x ^{2} \cdot x}

\frac{3 x ^{4} + x ^{2} - 5}{2 x ^{\frac{7}{2}}}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

\frac{( 4 x ^{3} - 2 x ) x ^{2} x - ( 2 x x + \frac{x ^{\frac{3}{2}}}{2} ) ( 1 - x ^{2} + x ^{4} )}{( x ^{2} x ) ^{2}} : \frac{3 x ^{4} + x ^{2} - 5}{2 x ^{\frac{7}{2}}}

= \frac{3 x ^{4} + x ^{2} - 5}{2 x ^{\frac{7}{2}}}

x \to 0 lim (\frac{0}{0})

\int \frac{2 x ^{2} + x - 1}{2 x ^{3} + x ^{2} - 5 x + 2} d x

\int \frac{x - 1}{x ^{2} - 4} d x

\int \frac{9}{x ^{2} - 16} d x

x \to 2 - lim (x^{3})