What is (\partial)/(\partial x)(1/(x-y)) ?

The answer to (\partial)/(\partial x)(1/(x-y)) is -1/((x-y)^2)

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(\partial)/(\partial x)(1/(x-y))

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

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

Solution steps

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

Treat

y

as a constant

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

What is (\partial)/(\partial x)(1/(x-y)) ?
The answer to (\partial)/(\partial x)(1/(x-y)) is -1/((x-y)^2)