derivative of (1-1/(x^2)× (2+x))

Lösung

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

\frac{4 + x}{x ^{3}} + 1

Schritte zur Lösung

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

Wende die Produktregel an:

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

f = 1 - \frac{1}{x ^{2}}, g = 2 + x

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

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

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

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

Vereinfache

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

= \frac{4 + x}{x ^{3}} + 1

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

e x p an d x (1 + x)

s im pl i f y (2 u^{2} + 2 v^{2}) (2 u^{2} + 2 v^{2} + 4)

e x p an d \frac{( 1 )}{( - 2 x ^{(2)} + 2 y ) ( 8 x + 2 y )}

e x p an d \frac{2 t ^{2} + t + 1}{( x - 3 ) ( 2 x ^{2} + 1 )}