Solve 1/x-y-3xy/x^3-y^3

$$\frac { 1 } { x - y } - \frac { 3 x y } { x ^ { 3 } - y ^ { 3 } }$$

Evaluate

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

Short Solution Steps

Factor x^{3}-y^{3}.

$$\frac{1}{x-y}-\frac{3xy}{\left(x-y\right)\left(x^{2}+xy+y^{2}\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-y and \left(x-y\right)\left(x^{2}+xy+y^{2}\right) is \left(x-y\right)\left(x^{2}+xy+y^{2}\right). Multiply \frac{1}{x-y} times \frac{x^{2}+xy+y^{2}}{x^{2}+xy+y^{2}}.

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

Since \frac{x^{2}+xy+y^{2}}{\left(x-y\right)\left(x^{2}+xy+y^{2}\right)} and \frac{3xy}{\left(x-y\right)\left(x^{2}+xy+y^{2}\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

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

Factor the expressions that are not already factored in \frac{x^{2}+y^{2}-2xy}{\left(x-y\right)\left(x^{2}+xy+y^{2}\right)}.

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

Cancel out x-y in both numerator and denominator.

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

Differentiate w.r.t. x

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