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

The answer to (\partial)/(\partial x)((2x-3y)/(2x+3y)) is (12y)/((2x+3y)^2)

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

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

\frac{12 y}{( 2 x + 3 y ) ^{2}}

Solution steps

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

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

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

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

Expand

2 (2 x + 3 y) - 2 (2 x - 3 y) : 12 y

= \frac{12 y}{( 2 x + 3 y ) ^{2}}

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

What is (\partial)/(\partial x)((2x-3y)/(2x+3y)) ?
The answer to (\partial)/(\partial x)((2x-3y)/(2x+3y)) is (12y)/((2x+3y)^2)