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

The answer to (\partial)/(\partial x)((3x-2y)/(4x^2+y)) is (-12x^2+16xy+3y)/((4x^2+y)^2)

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(\partial)/(\partial x)((3x-2y)/(4x^2+y))

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

\frac{- 12 x ^{2} + 16 x y + 3 y}{( 4 x ^{2} + y ) ^{2}}

Solution steps

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

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

\frac{\partial}{\partial x} (4 x^{2} + y) = 8 x

= \frac{3 ( 4 x ^{2} + y ) - 8 x ( 3 x - 2 y )}{( 4 x ^{2} + y ) ^{2}}

Expand

3 (4 x^{2} + y) - 8 x (3 x - 2 y) : - 12 x^{2} + 16 x y + 3 y

= \frac{- 12 x ^{2} + 16 x y + 3 y}{( 4 x ^{2} + y ) ^{2}}

\int_{0}^{l n (5)} e^{x} d x

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What is (\partial)/(\partial x)((3x-2y)/(4x^2+y)) ?
The answer to (\partial)/(\partial x)((3x-2y)/(4x^2+y)) is (-12x^2+16xy+3y)/((4x^2+y)^2)