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

Lösung

ableitung von

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

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

\frac{d x}{d x} = 1

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

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

Vereinfache

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

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

d er i v a t i v e f (x) = \frac{x}{1 - ln ( x - 7 )}

y^{^{'}} = \frac{( ln ^{2} ( x ) )}{3 x}

d er i v a t i v e y = x^{2} - 3 x

x \to - 3 - lim (h (x))

s l o p e in t erce pt (1.7) (6.1)