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

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

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

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

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

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

Solution steps

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

Apply the Quotient Rule:

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

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

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

\frac{d}{d x} ((1 + x^{2})^{2}) = 4 x (1 + x^{2})

= \frac{( - 2 x - 2 ) ( 1 + x ^{2} ) ^{2} - 4 x ( 1 + x ^{2} ) ( - x ^{2} - 2 x + 1 )}{( ( 1 + x ^{2} ) ^{2} ) ^{2}}

Simplify

\frac{( - 2 x - 2 ) ( 1 + x ^{2} ) ^{2} - 4 x ( 1 + x ^{2} ) ( - x ^{2} - 2 x + 1 )}{( ( 1 + x ^{2} ) ^{2} ) ^{2}} : - \frac{2 ( - x ^{3} - 3 x ^{2} + 3 x + 1 )}{( x ^{2} + 1 ) ^{3}}

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

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