What is (\partial)/(\partial x)(-(2x)/((1+x^2)^2)) ?

The answer to (\partial)/(\partial x)(-(2x)/((1+x^2)^2)) is -(2(-3x^2+1))/((1+x^2)^3)

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(\partial)/(\partial x)(-(2x)/((1+x^2)^2))

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

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

Solution steps

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

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

\frac{\partial}{\partial x} (x) = 1

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

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

Simplify

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

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

What is (\partial)/(\partial x)(-(2x)/((1+x^2)^2)) ?
The answer to (\partial)/(\partial x)(-(2x)/((1+x^2)^2)) is -(2(-3x^2+1))/((1+x^2)^3)