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

Lösung

ableitung von

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

\frac{- 4 x ^{3} - 10 x ^{2} - 4 x + 2}{( x ^{2} + 1 ) ^{4}}

Schritte zur Lösung

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

Wende die Produktregel an:

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

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

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

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

\frac{d}{d x} ((x^{2} + 1)^{- 3}) = - \frac{6 x}{( x ^{2} + 1 ) ^{4}}

= 2 (x + 1) (x^{2} + 1)^{- 3} + (- \frac{6 x}{( x ^{2} + 1 ) ^{4}}) (x + 1)^{2}

Vereinfache

2 (x + 1) (x^{2} + 1)^{- 3} + (- \frac{6 x}{( x ^{2} + 1 ) ^{4}}) (x + 1)^{2} : \frac{- 4 x ^{3} - 10 x ^{2} - 4 x + 2}{( x ^{2} + 1 ) ^{4}}

= \frac{- 4 x ^{3} - 10 x ^{2} - 4 x + 2}{( x ^{2} + 1 ) ^{4}}

x \to 0 + lim (\frac{x - 2}{∣ x ∣})

d er i v a t i v e T

y^{^{''}} - 2 y + y = \frac{e ^{x}}{1 + x ^{2}}

y^{^{'}} sin (x) + y cos (x) = 0, y (\frac{π}{4}) = 1

x \to 0 + lim ((\frac{1}{x})^{l n (x)})