What is (\partial)/(\partial y)((xy)/(5x+3y)) ?

The answer to (\partial)/(\partial y)((xy)/(5x+3y)) is (5x^2)/((5x+3y)^2)

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(\partial)/(\partial y)((xy)/(5x+3y))

\frac{\partial}{\partial y} (\frac{x y}{5 x + 3 y})

\frac{5 x ^{2}}{( 5 x + 3 y ) ^{2}}

Solution steps

\frac{\partial}{\partial y} (\frac{x y}{5 x + 3 y})

Treat

x

as a constant

Take the constant out:

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

= x \frac{\partial}{\partial y} (\frac{y}{5 x + 3 y})

Apply the Quotient Rule:

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

= x \frac{\frac{\partial}{\partial y} ( y ) ( 5 x + 3 y ) - \frac{\partial}{\partial y} ( 5 x + 3 y ) y}{( 5 x + 3 y ) ^{2}}

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

\frac{\partial}{\partial y} (5 x + 3 y) = 3

= x \frac{1 \cdot ( 5 x + 3 y ) - 3 y}{( 5 x + 3 y ) ^{2}}

Simplify

x \frac{1 \cdot ( 5 x + 3 y ) - 3 y}{( 5 x + 3 y ) ^{2}} : \frac{5 x ^{2}}{( 5 x + 3 y ) ^{2}}

= \frac{5 x ^{2}}{( 5 x + 3 y ) ^{2}}

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What is (\partial)/(\partial y)((xy)/(5x+3y)) ?
The answer to (\partial)/(\partial y)((xy)/(5x+3y)) is (5x^2)/((5x+3y)^2)