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

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

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

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

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

Solution steps

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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