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

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

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

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

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

Solution steps

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

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

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

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

Simplify

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

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

\frac{\partial}{\partial y} (x y e^{6 y})

\int \frac{1}{sin ( u )} d u

\int \frac{5 + 8 x}{1 + x ^{2}} d x

\int (3 sec^{2} (4 x) + 3 csc^{2} (4 x)) d x

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