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

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

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(\partial)/(\partial y)((x+y)/(y-x))

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

- \frac{2 x}{( y - x ) ^{2}}

Solution steps

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

Treat

x

as a constant

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

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\int 27 e^{3 θ} sin (3 θ) d θ

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