derivative of 1/(x+x^{-1})

Lösung

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

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

Schritte zur Lösung

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

Wende Exponentenregel an:

\frac{1}{a} = a^{- 1}

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

t an g e n t x - 1

\frac{\partial}{\partial y} (x^{2} \cdot sin (2 y))

d er i v a t i v e f (x) = x - 3 x

d er i v a t i v e f (x) = x^{2} + x - 9

x \to \frac{1}{8} lim (\frac{1}{x} + x^{- 1})