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

The answer to (\partial)/(\partial x)((3x-2y)/(x^2+3y)) is (-3x^2+4xy+9y)/((x^2+3y)^2)

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

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

\frac{- 3 x ^{2} + 4 x y + 9 y}{( x ^{2} + 3 y ) ^{2}}

Solution steps

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

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

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

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

Expand

3 (x^{2} + 3 y) - 2 x (3 x - 2 y) : - 3 x^{2} + 4 x y + 9 y

= \frac{- 3 x ^{2} + 4 x y + 9 y}{( x ^{2} + 3 y ) ^{2}}

\int \frac{cos ( x )}{sin ^{2} ( x ) - 36} d x

y^{^{''}} + 2 y^{^{'}} + 122 y = 0

\frac{\partial}{\partial x} (sin (- e^{y}))

\frac{d}{d x} (cos (\frac{3 x}{2}))

\frac{d}{d x} (9 x^{2} + 9)

What is (\partial)/(\partial x)((3x-2y)/(x^2+3y)) ?
The answer to (\partial)/(\partial x)((3x-2y)/(x^2+3y)) is (-3x^2+4xy+9y)/((x^2+3y)^2)