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

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

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

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

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

Solution steps

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

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

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

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

Simplify

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

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

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