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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

- \frac{1}{( x - y ) ^{2}} \frac{\partial}{\partial y} (x - y)

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

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

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

Simplify

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

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

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