derivative y=(x+1/x)(x-1/x+1)

Lösung

ableitung von

y = (x + \frac{1}{x}) (x - \frac{1}{x} + 1)

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

Schritte zur Lösung

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

Wende die Produktregel an:

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

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

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

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

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

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

Vereinfache

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

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