Solve 1/x-y-2y/x^2-y^2=

$$\frac { 1 } { x - y } - \frac { 2 y } { x ^ { 2 } - y ^ { 2 } } =$$

Evaluate

$$\frac{1}{x+y}$$

Short Solution Steps

Factor x^{2}-y^{2}.

$$\frac{1}{x-y}-\frac{2y}{\left(x+y\right)\left(x-y\right)}$$

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}.

$$\frac{x+y}{\left(x+y\right)\left(x-y\right)}-\frac{2y}{\left(x+y\right)\left(x-y\right)}$$

Since \frac{x+y}{\left(x+y\right)\left(x-y\right)} and \frac{2y}{\left(x+y\right)\left(x-y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x+y-2y}{\left(x+y\right)\left(x-y\right)}$$

Combine like terms in x+y-2y.

$$\frac{x-y}{\left(x+y\right)\left(x-y\right)}$$

Cancel out x-y in both numerator and denominator.

$$\frac{1}{x+y}$$

Differentiate w.r.t. x

$$-\frac{1}{\left(x+y\right)^{2}}$$