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Differentiate w.r.t. x
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\frac{1}{x-y}-\frac{2xy}{\left(x+y\right)\left(x-y\right)}
Factor x^{2}-y^{2}.
\frac{x+y}{\left(x+y\right)\left(x-y\right)}-\frac{2xy}{\left(x+y\right)\left(x-y\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-y and \left(x+y\right)\left(x-y\right) is \left(x+y\right)\left(x-y\right). Multiply \frac{1}{x-y} times \frac{x+y}{x+y}.
\frac{x+y-2xy}{\left(x+y\right)\left(x-y\right)}
Since \frac{x+y}{\left(x+y\right)\left(x-y\right)} and \frac{2xy}{\left(x+y\right)\left(x-y\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{x+y-2xy}{x^{2}-y^{2}}
Expand \left(x+y\right)\left(x-y\right).