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

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

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

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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