What is (\partial)/(\partial x)((5x-9y)/(5x+9y)) ?

The answer to (\partial)/(\partial x)((5x-9y)/(5x+9y)) is (90y)/((5x+9y)^2)

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(\partial)/(\partial x)((5x-9y)/(5x+9y))

\frac{\partial}{\partial x} (\frac{5 x - 9 y}{5 x + 9 y})

\frac{90 y}{( 5 x + 9 y ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{5 x - 9 y}{5 x + 9 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} ( 5 x - 9 y ) ( 5 x + 9 y ) - \frac{\partial}{\partial x} ( 5 x + 9 y ) ( 5 x - 9 y )}{( 5 x + 9 y ) ^{2}}

\frac{\partial}{\partial x} (5 x - 9 y) = 5

\frac{\partial}{\partial x} (5 x + 9 y) = 5

= \frac{5 ( 5 x + 9 y ) - 5 ( 5 x - 9 y )}{( 5 x + 9 y ) ^{2}}

Expand

5 (5 x + 9 y) - 5 (5 x - 9 y) : 90 y

= \frac{90 y}{( 5 x + 9 y ) ^{2}}

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What is (\partial)/(\partial x)((5x-9y)/(5x+9y)) ?
The answer to (\partial)/(\partial x)((5x-9y)/(5x+9y)) is (90y)/((5x+9y)^2)