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

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

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

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

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

Solution steps

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

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

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

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

Simplify

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

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

x \to 4 - lim (\frac{1}{x ^{2} - 3 x - 4})

\int \frac{( x - 1 )}{x ^{2}} e^{x} d x

f (x) = (6 x - 2)^{2} \cdot (6 x^{2} + 3)^{5}

y = (2 x - 9)^{4} (x^{2} + x + 1)^{5}

\frac{d}{d x} (x^{2} (\frac{2}{x ^{2}} + \frac{5}{x ^{3}}))

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