derivative of (2x^3/(2x^4-5x^3+x^2))

Lösung

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

- \frac{2 ( 2 x ^{2} - 1 )}{( 2 x ^{2} - 5 x + 1 ) ^{2}}

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

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

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

\frac{d}{d x} (2 x^{4} - 5 x^{3} + x^{2}) = 8 x^{3} - 15 x^{2} + 2 x

= 2 \cdot \frac{3 x ^{2} ( 2 x ^{4} - 5 x ^{3} + x ^{2} ) - ( 8 x ^{3} - 15 x ^{2} + 2 x ) x ^{3}}{( 2 x ^{4} - 5 x ^{3} + x ^{2} ) ^{2}}

Vereinfache

2 \cdot \frac{3 x ^{2} ( 2 x ^{4} - 5 x ^{3} + x ^{2} ) - ( 8 x ^{3} - 15 x ^{2} + 2 x ) x ^{3}}{( 2 x ^{4} - 5 x ^{3} + x ^{2} ) ^{2}} : - \frac{2 ( 2 x ^{2} - 1 )}{( 2 x ^{2} - 5 x + 1 ) ^{2}}

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

d er i v a t i v e f (x) = 4 e^{x}

d er i v a t i v e x (x - 10)

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

\int (x + 3)^{3} d x

x \to - 4 lim (x^{3} + 4 x^{2} - 8 x)