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

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

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(\partial)/(\partial x)(x/((x^2+y^2+z^2)))

\frac{\partial}{\partial x} (\frac{x}{( x ^{2} + y ^{2} + z ^{2} )})

\frac{- x ^{2} + y ^{2} + z ^{2}}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{x}{( x ^{2} + y ^{2} + z ^{2} )})

Treat

y, z

as constants

Apply the Quotient Rule:

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

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

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

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

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

1 \cdot (x^{2} + y^{2} + z^{2}) - 2 xx = - x^{2} + y^{2} + z^{2}

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

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