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

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

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(\partial)/(\partial y)((xy)/(x+y+z))

\frac{\partial}{\partial y} (\frac{x y}{x + y + z})

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

Solution steps

\frac{\partial}{\partial y} (\frac{x y}{x + y + z})

Treat

x, z

as constants

Take the constant out:

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

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

Simplify

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

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

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