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

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

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

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

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

Solution steps

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

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

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

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

Simplify

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

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

x \to 0 lim (x - \frac{1}{x - 1})

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\int_{0}^{1} \frac{3 x ^{3}}{4 x ^{4} + 9} d x

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