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

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

\frac{d}{d x} (x e^{- x} - e^{- x})

x \to 0 lim (\frac{1}{3 x})

x y^{^{'}} - y = x ln (x)

d er i v a t i v e 4^{x} x^{4}

x (x + 1) (\frac{d y}{d x}) + x y = 1