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

The derivative of-(2(-3x^2+1)/((1+x^2)^3)) is (24x(-x^2+1))/((x^2+1)^4)

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

The first derivative of-(2(-3x^2+1)/((1+x^2)^3)) is (24x(-x^2+1))/((x^2+1)^4)

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

\frac{24 x ( - x ^{2} + 1 )}{( x ^{2} + 1 ) ^{4}}

Solution steps

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

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Quotient Rule:

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

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

\frac{d}{d x} (- 3 x^{2} + 1) = - 6 x

\frac{d}{d x} ((1 + x^{2})^{3}) = 6 x (1 + x^{2})^{2}

= - 2 \cdot \frac{( - 6 x ) ( 1 + x ^{2} ) ^{3} - 6 x ( 1 + x ^{2} ) ^{2} ( - 3 x ^{2} + 1 )}{( ( 1 + x ^{2} ) ^{3} ) ^{2}}

Simplify

- 2 \cdot \frac{( - 6 x ) ( 1 + x ^{2} ) ^{3} - 6 x ( 1 + x ^{2} ) ^{2} ( - 3 x ^{2} + 1 )}{( ( 1 + x ^{2} ) ^{3} ) ^{2}} : \frac{24 x ( - x ^{2} + 1 )}{( x ^{2} + 1 ) ^{4}}

= \frac{24 x ( - x ^{2} + 1 )}{( x ^{2} + 1 ) ^{4}}

\frac{x ^{4} - x + 1}{x ^{4} + x + 1}

\frac{\partial}{\partial y} (x^{2} + x y + 3 x + 2 y + 5)

\frac{d}{d x} (\frac{1}{x ^{2} + 16})

\int_{0}^{4} 4 x + 9 d x

\int_{- 9}^{9} 81 - x^{2} d x

What is the derivative of-(2(-3x^2+1)/((1+x^2)^3)) ?
The derivative of-(2(-3x^2+1)/((1+x^2)^3)) is (24x(-x^2+1))/((x^2+1)^4)
What is the first derivative of-(2(-3x^2+1)/((1+x^2)^3)) ?
The first derivative of-(2(-3x^2+1)/((1+x^2)^3)) is (24x(-x^2+1))/((x^2+1)^4)