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

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

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

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

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

Solution steps

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

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\frac{d}{d x} ((ln (x))^{4})

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