What is (\partial)/(\partial y)(y/(2x+3y)) ?

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

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(\partial)/(\partial y)(y/(2x+3y))

\frac{\partial}{\partial y} (\frac{y}{2 x + 3 y})

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

Solution steps

\frac{\partial}{\partial y} (\frac{y}{2 x + 3 y})

Treat

x

as a constant

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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