Solve 5/x+y-4/x-y-8x/y^2-x^2

$$\frac{ 5 }{ x+y } - \frac{ 4 }{ x-y } - \frac{ 8x }{ { y }^{ 2 } - { x }^{ 2 } }$$

Evaluate

$$\frac{9}{x+y}$$

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{5}{x+y} times \frac{x-y}{x-y}. Multiply \frac{4}{x-y} times \frac{x+y}{x+y}.

$$\frac{5\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}-\frac{4\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}-\frac{8x}{y^{2}-x^{2}}$$

Since \frac{5\left(x-y\right)}{\left(x+y\right)\left(x-y\right)} and \frac{4\left(x+y\right)}{\left(x+y\right)\left(x-y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\left(x-y\right)-4\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}-\frac{8x}{y^{2}-x^{2}}$$

Do the multiplications in 5\left(x-y\right)-4\left(x+y\right).

$$\frac{5x-5y-4x-4y}{\left(x+y\right)\left(x-y\right)}-\frac{8x}{y^{2}-x^{2}}$$

Combine like terms in 5x-5y-4x-4y.

$$\frac{x-9y}{\left(x+y\right)\left(x-y\right)}-\frac{8x}{y^{2}-x^{2}}$$

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

$$\frac{x-9y}{\left(x+y\right)\left(x-y\right)}-\frac{8x}{\left(x+y\right)\left(-x+y\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+y\right)\left(x-y\right) and \left(x+y\right)\left(-x+y\right) is \left(x+y\right)\left(-x+y\right). Multiply \frac{x-9y}{\left(x+y\right)\left(x-y\right)} times \frac{-1}{-1}.

$$\frac{-\left(x-9y\right)}{\left(x+y\right)\left(-x+y\right)}-\frac{8x}{\left(x+y\right)\left(-x+y\right)}$$

Since \frac{-\left(x-9y\right)}{\left(x+y\right)\left(-x+y\right)} and \frac{8x}{\left(x+y\right)\left(-x+y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\left(x-9y\right)-8x}{\left(x+y\right)\left(-x+y\right)}$$

Do the multiplications in -\left(x-9y\right)-8x.

$$\frac{-x+9y-8x}{\left(x+y\right)\left(-x+y\right)}$$

Combine like terms in -x+9y-8x.

$$\frac{-9x+9y}{\left(x+y\right)\left(-x+y\right)}$$

Factor the expressions that are not already factored in \frac{-9x+9y}{\left(x+y\right)\left(-x+y\right)}.

$$\frac{9\left(-x+y\right)}{\left(x+y\right)\left(-x+y\right)}$$

Cancel out -x+y in both numerator and denominator.

$$\frac{9}{x+y}$$

Differentiate w.r.t. x

$$-\frac{9}{\left(x+y\right)^{2}}$$