What is (\partial)/(\partial x)((x^2+y^2)^{-15/2}x) ?

The answer to (\partial)/(\partial x)((x^2+y^2)^{-15/2}x) is (-14x^2+y^2)/((x^2+y^2)^{17/2)}

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(\partial)/(\partial x)((x^2+y^2)^{-15/2}x)

\frac{\partial}{\partial x} ((x^{2} + y^{2})^{- \frac{15}{2}} x)

\frac{- 14 x ^{2} + y ^{2}}{( x ^{2} + y ^{2} ) ^{\frac{17}{2}}}

Solution steps

\frac{\partial}{\partial x} ((x^{2} + y^{2})^{- \frac{15}{2}} x)

Treat

y

as a constant

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = (x^{2} + y^{2})^{- \frac{15}{2}}, g = x

= \frac{\partial}{\partial x} ((x^{2} + y^{2})^{- \frac{15}{2}}) x + \frac{\partial}{\partial x} (x) (x^{2} + y^{2})^{- \frac{15}{2}}

\frac{\partial}{\partial x} ((x^{2} + y^{2})^{- \frac{15}{2}}) = - \frac{15 x}{( x ^{2} + y ^{2} ) ^{\frac{17}{2}}}

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

= (- \frac{15 x}{( x ^{2} + y ^{2} ) ^{\frac{17}{2}}}) x + 1 \cdot (x^{2} + y^{2})^{- \frac{15}{2}}

Simplify

(- \frac{15 x}{( x ^{2} + y ^{2} ) ^{\frac{17}{2}}}) x + 1 \cdot (x^{2} + y^{2})^{- \frac{15}{2}} : \frac{- 14 x ^{2} + y ^{2}}{( x ^{2} + y ^{2} ) ^{\frac{17}{2}}}

= \frac{- 14 x ^{2} + y ^{2}}{( x ^{2} + y ^{2} ) ^{\frac{17}{2}}}

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What is (\partial)/(\partial x)((x^2+y^2)^{-15/2}x) ?
The answer to (\partial)/(\partial x)((x^2+y^2)^{-15/2}x) is (-14x^2+y^2)/((x^2+y^2)^{17/2)}