derivative-2x^3(x^2+1)^{-2}

Lösung

ableitung von

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{3}, g = (x^{2} + 1)^{- 2}

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

\frac{d}{d x} (x^{3}) = 3 x^{2}

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

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

Vereinfache

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

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

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

t an g e n t y = 3 x^{2} + 4 x, (- 2, 4)

\frac{\partial}{\partial x} (2 x y^{7} + x^{8} y^{7})

d er i v a t i v e f (x) = x^{2} + 9

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