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

The answer to (\partial)/(\partial x)((x-3y)/(2xy+3)) is (6y^2+3)/((2xy+3)^2)

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

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

\frac{6 y ^{2} + 3}{( 2 x y + 3 ) ^{2}}

Solution steps

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

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

\frac{\partial}{\partial x} (x - 3 y) = 1

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

= \frac{1 \cdot ( 2 x y + 3 ) - 2 y ( x - 3 y )}{( 2 x y + 3 ) ^{2}}

Simplify

\frac{1 \cdot ( 2 x y + 3 ) - 2 y ( x - 3 y )}{( 2 x y + 3 ) ^{2}} : \frac{6 y ^{2} + 3}{( 2 x y + 3 ) ^{2}}

= \frac{6 y ^{2} + 3}{( 2 x y + 3 ) ^{2}}

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What is (\partial)/(\partial x)((x-3y)/(2xy+3)) ?
The answer to (\partial)/(\partial x)((x-3y)/(2xy+3)) is (6y^2+3)/((2xy+3)^2)