What is (\partial)/(\partial x)((4x-2y)/(4x+2y)) ?

The answer to (\partial)/(\partial x)((4x-2y)/(4x+2y)) is (4y)/((2x+y)^2)

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(\partial)/(\partial x)((4x-2y)/(4x+2y))

\frac{\partial}{\partial x} (\frac{4 x - 2 y}{4 x + 2 y})

\frac{4 y}{( 2 x + y ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{4 x - 2 y}{4 x + 2 y})

Treat

y

as a constant

Apply the Quotient Rule:

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

= \frac{\frac{\partial}{\partial x} ( 4 x - 2 y ) ( 4 x + 2 y ) - \frac{\partial}{\partial x} ( 4 x + 2 y ) ( 4 x - 2 y )}{( 4 x + 2 y ) ^{2}}

\frac{\partial}{\partial x} (4 x - 2 y) = 4

\frac{\partial}{\partial x} (4 x + 2 y) = 4

= \frac{4 ( 4 x + 2 y ) - 4 ( 4 x - 2 y )}{( 4 x + 2 y ) ^{2}}

Simplify

\frac{4 ( 4 x + 2 y ) - 4 ( 4 x - 2 y )}{( 4 x + 2 y ) ^{2}} : \frac{4 y}{( 2 x + y ) ^{2}}

= \frac{4 y}{( 2 x + y ) ^{2}}

What is (\partial)/(\partial x)((4x-2y)/(4x+2y)) ?
The answer to (\partial)/(\partial x)((4x-2y)/(4x+2y)) is (4y)/((2x+y)^2)