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, add them by adding 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

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