Solve x/x-y-y/x+y

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-y and x+y is \left(x+y\right)\left(x-y\right). Multiply \frac{x}{x-y} times \frac{x+y}{x+y}. Multiply \frac{y}{x+y} times \frac{x-y}{x-y}.

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

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

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

Do the multiplications in x\left(x+y\right)-y\left(x-y\right).

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

Combine like terms in x^{2}+xy-yx+y^{2}.

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

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

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

Differentiate w.r.t. x

$$-4\times \left(\frac{y}{x^{2}-y^{2}}\right)^{2}x$$