derivative of (250000x^2/((2x+1)^2))

Lösung

\frac{d}{d x} (\frac{250000 x ^{2}}{( 2 x + 1 ) ^{2}})

\frac{500000 x}{( 2 x + 1 ) ^{3}}

Schritte zur Lösung

\frac{d}{d x} (\frac{250000 x ^{2}}{( 2 x + 1 ) ^{2}})

Entferne die Konstante:

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

= 250000 \frac{d}{d x} (\frac{x ^{2}}{( 2 x + 1 ) ^{2}})

Wende die Quotientenregel an:

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

= 250000 \cdot \frac{\frac{d}{d x} ( x ^{2} ) ( 2 x + 1 ) ^{2} - \frac{d}{d x} ( ( 2 x + 1 ) ^{2} ) x ^{2}}{( ( 2 x + 1 ) ^{2} ) ^{2}}

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

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

= 250000 \cdot \frac{2 x ( 2 x + 1 ) ^{2} - 4 ( 2 x + 1 ) x ^{2}}{( ( 2 x + 1 ) ^{2} ) ^{2}}

Vereinfache

250000 \cdot \frac{2 x ( 2 x + 1 ) ^{2} - 4 ( 2 x + 1 ) x ^{2}}{( ( 2 x + 1 ) ^{2} ) ^{2}} : \frac{500000 x}{( 2 x + 1 ) ^{3}}

= \frac{500000 x}{( 2 x + 1 ) ^{3}}