What is (\partial)/(\partial x)(((2x+3y))/(4x+5y)) ?

The answer to (\partial)/(\partial x)(((2x+3y))/(4x+5y)) is -(2y)/((4x+5y)^2)

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(\partial)/(\partial x)(((2x+3y))/(4x+5y))

\frac{\partial}{\partial x} (\frac{( 2 x + 3 y )}{4 x + 5 y})

- \frac{2 y}{( 4 x + 5 y ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{( 2 x + 3 y )}{4 x + 5 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 ) ( 4 x + 5 y ) - \frac{\partial}{\partial x} ( 4 x + 5 y ) ( 2 x + 3 y )}{( 4 x + 5 y ) ^{2}}

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

\frac{\partial}{\partial x} (4 x + 5 y) = 4

= \frac{2 ( 4 x + 5 y ) - 4 ( 2 x + 3 y )}{( 4 x + 5 y ) ^{2}}

Simplify

\frac{2 ( 4 x + 5 y ) - 4 ( 2 x + 3 y )}{( 4 x + 5 y ) ^{2}} : - \frac{2 y}{( 4 x + 5 y ) ^{2}}

= - \frac{2 y}{( 4 x + 5 y ) ^{2}}

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What is (\partial)/(\partial x)(((2x+3y))/(4x+5y)) ?
The answer to (\partial)/(\partial x)(((2x+3y))/(4x+5y)) is -(2y)/((4x+5y)^2)