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

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

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

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

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

Solution steps

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

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

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

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

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

Simplify

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

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

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