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

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

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

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

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

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

Solution steps

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

Apply the Quotient Rule:

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

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

\frac{d x}{d x} = 1

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

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

Simplify

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

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

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