What is the derivative of 1/(x+x^{-1}) ?

The derivative of 1/(x+x^{-1}) is -(x^2-1)/((x^2+1)^2)

What is the first derivative of 1/(x+x^{-1}) ?

The first derivative of 1/(x+x^{-1}) is -(x^2-1)/((x^2+1)^2)

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

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

Solution steps

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

Apply exponent rule:

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

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

Apply the chain rule:

- \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}})

Simplify

- \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}}

x - 1

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

f (x) = x - 3 x

f (x) = x^{2} + x - 9

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

What is the derivative of 1/(x+x^{-1}) ?
The derivative of 1/(x+x^{-1}) is -(x^2-1)/((x^2+1)^2)
What is the first derivative of 1/(x+x^{-1}) ?
The first derivative of 1/(x+x^{-1}) is -(x^2-1)/((x^2+1)^2)