What is (\partial)/(\partial x)(xy((8-xy)/(x+y))) ?

The answer to (\partial)/(\partial x)(xy((8-xy)/(x+y))) is (y(-x^2y-2xy^2+8y))/((x+y)^2)

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(\partial)/(\partial x)(xy((8-xy)/(x+y)))

\frac{\partial}{\partial x} (x y (\frac{8 - x y}{x + y}))

\frac{y ( - x ^{2} y - 2 x y ^{2} + 8 y )}{( x + y ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (x y (\frac{8 - 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{8 - x y}{x + y})

Apply the Product Rule:

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

f = x, g = \frac{8 - x y}{x + y}

= y (\frac{\partial}{\partial x} (x) \frac{8 - x y}{x + y} + \frac{\partial}{\partial x} (\frac{8 - x y}{x + y}) x)

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

\frac{\partial}{\partial x} (\frac{8 - x y}{x + y}) = \frac{- y ^{2} - 8}{( x + y ) ^{2}}

= y (1 \cdot \frac{8 - x y}{x + y} + \frac{- y ^{2} - 8}{( x + y ) ^{2}} x)

Simplify

y (1 \cdot \frac{8 - x y}{x + y} + \frac{- y ^{2} - 8}{( x + y ) ^{2}} x) : \frac{y ( - x ^{2} y - 2 x y ^{2} + 8 y )}{( x + y ) ^{2}}

= \frac{y ( - x ^{2} y - 2 x y ^{2} + 8 y )}{( x + y ) ^{2}}

\int_{0}^{\frac{π}{7}} cos (4 x) d x

\int x^{2} + 1 d x

\frac{d}{d x} (e^{2 x - x^{2}})

n = 0 \sum \infty \frac{6 ^{n}}{n + 1}

x \to - \frac{7 π}{2} + lim (sec (x))

What is (\partial)/(\partial x)(xy((8-xy)/(x+y))) ?
The answer to (\partial)/(\partial x)(xy((8-xy)/(x+y))) is (y(-x^2y-2xy^2+8y))/((x+y)^2)