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

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

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

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

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

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

Solution steps

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

Simplify

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

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

\int \frac{sin ^{2} ( x )}{sin ( x ) + cos ( x )} d x

t \to 0 lim (sin (t))

\frac{d}{d x} (2 x^{\frac{5}{3}} - 5 x^{\frac{4}{5}} + 7 x^{\frac{3}{2}} - 8 x + 17)

\int arcsin^{2} (x) d x

f (x) = \frac{x ^{2}}{x - 11}

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