Solve 1/p-q-3pq/p^3-q^3

$$\frac { 1 } { p - q } - \frac { 3 p q } { p ^ { 3 } - q ^ { 3 } }$$

Evaluate

$$\frac{p-q}{p^{2}+pq+q^{2}}$$

Short Solution Steps

Factor p^{3}-q^{3}.

$$\frac{1}{p-q}-\frac{3pq}{\left(p-q\right)\left(p^{2}+pq+q^{2}\right)}$$

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

$$\frac{p^{2}+pq+q^{2}}{\left(p-q\right)\left(p^{2}+pq+q^{2}\right)}-\frac{3pq}{\left(p-q\right)\left(p^{2}+pq+q^{2}\right)}$$

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

$$\frac{p^{2}+pq+q^{2}-3pq}{\left(p-q\right)\left(p^{2}+pq+q^{2}\right)}$$

Combine like terms in p^{2}+pq+q^{2}-3pq.

$$\frac{p^{2}+q^{2}-2pq}{\left(p-q\right)\left(p^{2}+pq+q^{2}\right)}$$

Factor the expressions that are not already factored in \frac{p^{2}+q^{2}-2pq}{\left(p-q\right)\left(p^{2}+pq+q^{2}\right)}.

$$\frac{\left(p-q\right)^{2}}{\left(p-q\right)\left(p^{2}+pq+q^{2}\right)}$$

Cancel out p-q in both numerator and denominator.

$$\frac{p-q}{p^{2}+pq+q^{2}}$$

Differentiate w.r.t. p

$$\frac{2q^{2}+2pq-p^{2}}{\left(p^{2}+pq+q^{2}\right)^{2}}$$