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

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

Evaluate

$$-\frac{4y^{3}}{x^{4}-y^{4}}$$

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{1}{x+y} times \frac{x-y}{x-y}. Multiply \frac{1}{x-y} times \frac{x+y}{x+y}.

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

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

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

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

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

Combine like terms in x-y-x-y.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+y\right)\left(x-y\right) and x^{2}+y^{2} is \left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right). Multiply \frac{-2y}{\left(x+y\right)\left(x-y\right)} times \frac{x^{2}+y^{2}}{x^{2}+y^{2}}. Multiply \frac{2y}{x^{2}+y^{2}} times \frac{\left(x+y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}.

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

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

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

Do the multiplications in -2y\left(x^{2}+y^{2}\right)+2y\left(x+y\right)\left(x-y\right).

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

Combine like terms in -2yx^{2}-2y^{3}+2yx^{2}-2y^{2}x+2y^{2}x-2y^{3}.

$$\frac{-4y^{3}}{\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)}$$

Expand \left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right).

$$\frac{-4y^{3}}{x^{4}-y^{4}}$$

Differentiate w.r.t. x

$$\frac{16\left(xy\right)^{3}}{\left(x^{4}-y^{4}\right)^{2}}$$