What is (\partial)/(\partial y)((3x-4y)/(3x+4y)) ?

The answer to (\partial)/(\partial y)((3x-4y)/(3x+4y)) is -(24x)/((3x+4y)^2)

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(\partial)/(\partial y)((3x-4y)/(3x+4y))

\frac{\partial}{\partial y} (\frac{3 x - 4 y}{3 x + 4 y})

- \frac{24 x}{( 3 x + 4 y ) ^{2}}

Solution steps

\frac{\partial}{\partial y} (\frac{3 x - 4 y}{3 x + 4 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 - 4 y ) ( 3 x + 4 y ) - \frac{\partial}{\partial y} ( 3 x + 4 y ) ( 3 x - 4 y )}{( 3 x + 4 y ) ^{2}}

\frac{\partial}{\partial y} (3 x - 4 y) = - 4

\frac{\partial}{\partial y} (3 x + 4 y) = 4

= \frac{( - 4 ) ( 3 x + 4 y ) - 4 ( 3 x - 4 y )}{( 3 x + 4 y ) ^{2}}

Simplify

\frac{( - 4 ) ( 3 x + 4 y ) - 4 ( 3 x - 4 y )}{( 3 x + 4 y ) ^{2}} : - \frac{24 x}{( 3 x + 4 y ) ^{2}}

= - \frac{24 x}{( 3 x + 4 y ) ^{2}}

What is (\partial)/(\partial y)((3x-4y)/(3x+4y)) ?
The answer to (\partial)/(\partial y)((3x-4y)/(3x+4y)) is -(24x)/((3x+4y)^2)