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

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

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

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

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

Solution steps

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

Treat

y

as a constant

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

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

\frac{\partial}{\partial x} ((x^{2} + y^{2})^{2}) = 4 x (x^{2} + y^{2})

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

Simplify

\frac{1 \cdot ( x ^{2} + y ^{2} ) ^{2} - 4 x ( x ^{2} + y ^{2} ) x}{( ( x ^{2} + y ^{2} ) ^{2} ) ^{2}} : \frac{- 3 x ^{2} + y ^{2}}{( x ^{2} + y ^{2} ) ^{3}}

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

\int x^{2} 2 - x^{3} d x

cos (x), at \frac{π}{2}

\frac{\partial}{\partial y} ((x - y) (1 - x y))

x y^{^{'}} + 2 y = \frac{4 e ^{3 x}}{x}

f (x) = 2 x^{3} - 21 x^{2} + 72 x - 5

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