Solve 8x^2y^2/x^2-y^2:(x/x-y-x/x+y

$$\frac { 8 x ^ { 2 } y ^ { 2 } } { x ^ { 2 } - y ^ { 2 } } : ( \frac { x } { x - y } - \frac { x } { x + y }$$

Evaluate

$$4xy$$

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

$$\frac{\frac{8x^{2}y^{2}}{x^{2}-y^{2}}}{\frac{x\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}-\frac{x\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}}$$

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

$$\frac{\frac{8x^{2}y^{2}}{x^{2}-y^{2}}}{\frac{x\left(x+y\right)-x\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}}$$

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

$$\frac{\frac{8x^{2}y^{2}}{x^{2}-y^{2}}}{\frac{x^{2}+xy-x^{2}+xy}{\left(x+y\right)\left(x-y\right)}}$$

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

$$\frac{\frac{8x^{2}y^{2}}{x^{2}-y^{2}}}{\frac{2xy}{\left(x+y\right)\left(x-y\right)}}$$

Divide \frac{8x^{2}y^{2}}{x^{2}-y^{2}} by \frac{2xy}{\left(x+y\right)\left(x-y\right)} by multiplying \frac{8x^{2}y^{2}}{x^{2}-y^{2}} by the reciprocal of \frac{2xy}{\left(x+y\right)\left(x-y\right)}.

$$\frac{8x^{2}y^{2}\left(x+y\right)\left(x-y\right)}{\left(x^{2}-y^{2}\right)\times 2xy}$$

Cancel out 2xy in both numerator and denominator.

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

Factor the expressions that are not already factored.

$$\frac{4xy\left(x+y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}$$

Cancel out \left(x+y\right)\left(x-y\right) in both numerator and denominator.

$$4xy$$

Differentiate w.r.t. x

$$4y$$