derivative f(x)=(x^3-8x^2+5)/(x^2)

Lösung

ableitung von

f (x) = \frac{x ^{3} - 8 x ^{2} + 5}{x ^{2}}

\frac{x ^{3} - 10}{x ^{3}}

Schritte zur Lösung

\frac{d}{d x} (\frac{x ^{3} - 8 x ^{2} + 5}{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} - 8 x ^{2} + 5 ) x ^{2} - \frac{d}{d x} ( x ^{2} ) ( x ^{3} - 8 x ^{2} + 5 )}{( x ^{2} ) ^{2}}

\frac{d}{d x} (x^{3} - 8 x^{2} + 5) = 3 x^{2} - 16 x

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

= \frac{( 3 x ^{2} - 16 x ) x ^{2} - 2 x ( x ^{3} - 8 x ^{2} + 5 )}{( x ^{2} ) ^{2}}

Vereinfache

\frac{( 3 x ^{2} - 16 x ) x ^{2} - 2 x ( x ^{3} - 8 x ^{2} + 5 )}{( x ^{2} ) ^{2}} : \frac{x ^{3} - 10}{x ^{3}}

= \frac{x ^{3} - 10}{x ^{3}}

d er i v a t i v e 3 e^{y}

\frac{d}{d x} (- 2 x^{2} + \frac{5}{3} x + 11)

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

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

x \to 9 lim (\frac{∣ x - 9 ∣}{x - 9})