What is (\partial)/(\partial x)((xy)/(6x+7y)) ?

The answer to (\partial)/(\partial x)((xy)/(6x+7y)) is (7y^2)/((6x+7y)^2)

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(\partial)/(\partial x)((xy)/(6x+7y))

\frac{\partial}{\partial x} (\frac{x y}{6 x + 7 y})

\frac{7 y ^{2}}{( 6 x + 7 y ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{x y}{6 x + 7 y})

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= y \frac{\partial}{\partial x} (\frac{x}{6 x + 7 y})

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= y \frac{\frac{\partial}{\partial x} ( x ) ( 6 x + 7 y ) - \frac{\partial}{\partial x} ( 6 x + 7 y ) x}{( 6 x + 7 y ) ^{2}}

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

\frac{\partial}{\partial x} (6 x + 7 y) = 6

= y \frac{1 \cdot ( 6 x + 7 y ) - 6 x}{( 6 x + 7 y ) ^{2}}

Simplify

y \frac{1 \cdot ( 6 x + 7 y ) - 6 x}{( 6 x + 7 y ) ^{2}} : \frac{7 y ^{2}}{( 6 x + 7 y ) ^{2}}

= \frac{7 y ^{2}}{( 6 x + 7 y ) ^{2}}

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What is (\partial)/(\partial x)((xy)/(6x+7y)) ?
The answer to (\partial)/(\partial x)((xy)/(6x+7y)) is (7y^2)/((6x+7y)^2)