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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = \frac{2 - x y}{x + y}

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

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

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

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

Simplify

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

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

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

\int \frac{x ^{2} - 36}{x ^{2}} d x

\int \frac{1}{16 + x ^{2}} d x

x \to 0 + lim (- \frac{3 ln ( x )}{x})

y = 4 sec (x) - 8 cos (x)

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