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

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

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

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

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

Solution steps

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

Treat

x, z

as constants

Apply the chain rule:

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

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

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

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

Simplify

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

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

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