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

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

(x^{2} + 1) y^{^{'}} + 8 x (y - 1) = 0

d er i v a t i v e x^{4} - 8 x^{3}

\frac{\partial}{\partial x} (\frac{3}{3 x + 2 y})

\frac{d y}{d x} = \frac{x y + x ^{2} + y ^{2}}{x ^{2}}

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