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

Lösung

ableitung von

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

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

Schritte zur Lösung

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

Wende die Produktregel an:

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

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

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

Vereinfache

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

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

\frac{d}{d x} (\frac{cos ( x ^{3} )}{e ^{x}})

f (x) = x^{2} - 4

d er i v a t i v e x^{3} - x

t a y l or \frac{3}{1 - x}

s l o p e (- 4, - 4), (5, - 7)