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

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

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

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

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

Solution steps

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

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

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

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

Simplify

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

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

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