derivative (1-t)(1+t^2)^{-1}

Lösung

ableitung von

(1 - t) (1 + t^{2})^{- 1}

\frac{t ^{2} - 2 t - 1}{( t ^{2} + 1 ) ^{2}}

Schritte zur Lösung

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

Wende die Produktregel an:

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

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

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

\frac{d}{d t} (1 - t) = - 1

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

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

Vereinfache

(- 1) (1 + t^{2})^{- 1} + (- \frac{2 t}{( 1 + t ^{2} ) ^{2}}) (1 - t) : \frac{t ^{2} - 2 t - 1}{( t ^{2} + 1 ) ^{2}}

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