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

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

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

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

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

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

Solution steps

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

Apply the Quotient Rule:

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

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

\frac{d}{d x} (x^{4} + 3 x^{2}) = 4 x^{3} + 6 x

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

= \frac{( 4 x ^{3} + 6 x ) ( x ^{2} + 1 ) ^{2} - 4 x ( x ^{2} + 1 ) ( x ^{4} + 3 x ^{2} )}{( ( x ^{2} + 1 ) ^{2} ) ^{2}}

Simplify

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

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

\frac{\partial}{\partial x} (A)

x \to 1 lim (\frac{ln ( x )}{x})

- \frac{2}{x}

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

3 x^{2} + 8 x + 2

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