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

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

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

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

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

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d y}{d x} - 2 x y = 8 x

x \to 1 - lim (∣ x - 1 ∣)

\int \frac{4 x ^{3}}{e ^{2 x^{2}}} d x

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

f (x) = x^{2} + 2 x + 1, at x = - 3

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