What is (\partial)/(\partial x)((x^2+y^2+z^2)^{-1/4}) ?

The answer to (\partial)/(\partial x)((x^2+y^2+z^2)^{-1/4}) is -x/(2(x^2+y^2+z^2)^{5/4)}

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(\partial)/(\partial x)((x^2+y^2+z^2)^{-1/4})

\frac{\partial}{\partial x} ((x^{2} + y^{2} + z^{2})^{- \frac{1}{4}})

- \frac{x}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{4}}}

Solution steps

\frac{\partial}{\partial x} ((x^{2} + y^{2} + z^{2})^{- \frac{1}{4}})

Treat

y, z

as constants

Apply the chain rule:

- \frac{1}{4 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{4}}} \frac{\partial}{\partial x} (x^{2} + y^{2} + z^{2})

= - \frac{1}{4 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{4}}} \frac{\partial}{\partial x} (x^{2} + y^{2} + z^{2})

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

= - \frac{1}{4 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{4}}} \cdot 2 x

Simplify

- \frac{1}{4 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{4}}} \cdot 2 x : - \frac{x}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{4}}}

= - \frac{x}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{4}}}

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What is (\partial)/(\partial x)((x^2+y^2+z^2)^{-1/4}) ?
The answer to (\partial)/(\partial x)((x^2+y^2+z^2)^{-1/4}) is -x/(2(x^2+y^2+z^2)^{5/4)}