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

The answer to (\partial)/(\partial x)((4x-2y)/(3x^2+2y)) is (-12x^2+12xy+8y)/((3x^2+2y)^2)

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

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

\frac{- 12 x ^{2} + 12 x y + 8 y}{( 3 x ^{2} + 2 y ) ^{2}}

Solution steps

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

\frac{\partial}{\partial x} (4 x - 2 y) = 4

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

= \frac{4 ( 3 x ^{2} + 2 y ) - 6 x ( 4 x - 2 y )}{( 3 x ^{2} + 2 y ) ^{2}}

Expand

4 (3 x^{2} + 2 y) - 6 x (4 x - 2 y) : - 12 x^{2} + 12 x y + 8 y

= \frac{- 12 x ^{2} + 12 x y + 8 y}{( 3 x ^{2} + 2 y ) ^{2}}

\frac{d y}{d x} = \frac{- 2 x y + 1}{x ^{2} + 2 y}

- \frac{7}{2} x^{2} + 5 x + 5

\int x^{b - 1} e^{x^{b}} d x

\int 200 - 4 t d t

\frac{d}{d x} (lo g_{10} (\frac{2 x}{1 + x ^{2}}))

What is (\partial)/(\partial x)((4x-2y)/(3x^2+2y)) ?
The answer to (\partial)/(\partial x)((4x-2y)/(3x^2+2y)) is (-12x^2+12xy+8y)/((3x^2+2y)^2)