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

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

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

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

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

Vereinfache

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

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

x \to 1 lim (2 x - x^{2} - x^{3})

\frac{\partial}{\partial x} (2 x^{2} - 4)

\frac{\partial}{\partial x} (1 + x y)

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

t an g e n t \frac{2}{x - 2}