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

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

Evaluate

$$\frac{y+x-2xy}{x^{2}-y^{2}}$$

Short Solution Steps

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

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

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}.

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

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.

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

Expand \left(x+y\right)\left(x-y\right).

$$\frac{x+y-2xy}{x^{2}-y^{2}}$$

Differentiate w.r.t. x

$$\frac{2yx^{2}-y^{2}+2y^{3}-2xy-x^{2}}{\left(x^{2}-y^{2}\right)^{2}}$$