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

The derivative of (x+1/x (x-1/x)) is 2x+2/(x^3)

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

The first derivative of (x+1/x (x-1/x)) is 2x+2/(x^3)

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

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

Solution steps

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

Apply the Product Rule:

(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})

Simplify

(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}}

What is the derivative of (x+1/x (x-1/x)) ?
The derivative of (x+1/x (x-1/x)) is 2x+2/(x^3)
What is the first derivative of (x+1/x (x-1/x)) ?
The first derivative of (x+1/x (x-1/x)) is 2x+2/(x^3)