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

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

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

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

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

Solution steps

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

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

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

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

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

Simplify

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

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

\int \frac{2 + ln ( x )}{x ln ( x )} d x

y = 3 x^{2} + 3 x

\int_{2}^{\infty} \frac{9}{x ^{2}} d x

\int 5 x ln (x^{3}) d x

e^{x}, e^{- 2 x}, x = ln (5)

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