derivative f(x)=(x^3-x^2-1)/(x^2)

Lösung

ableitung von

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

\frac{x ^{3} + 2}{x ^{3}}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

= \frac{x ^{3} + 2}{x ^{3}}

x \to - 2 lim (2 + x)

x \to 0 lim (\frac{x - 1}{x})

\int \frac{4}{x ^{2} - x - 6} d x

\int 2 x + 2 d x

d er i v a t i v e 6 + 5 x