Solve 3/a+b+2/a-b-1/a^2-b^2=

$$\frac { 3 } { a + b } + \frac { 2 } { a - b } - \frac { 1 } { a ^ { 2 } - b ^ { 2 } } =$$

Evaluate

$$\frac{5a-b-1}{a^{2}-b^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+b and a-b is \left(a+b\right)\left(a-b\right). Multiply \frac{3}{a+b} times \frac{a-b}{a-b}. Multiply \frac{2}{a-b} times \frac{a+b}{a+b}.

$$\frac{3\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}+\frac{2\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{1}{a^{2}-b^{2}}$$

Since \frac{3\left(a-b\right)}{\left(a+b\right)\left(a-b\right)} and \frac{2\left(a+b\right)}{\left(a+b\right)\left(a-b\right)} have the same denominator, add them by adding their numerators.

$$\frac{3\left(a-b\right)+2\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{1}{a^{2}-b^{2}}$$

Do the multiplications in 3\left(a-b\right)+2\left(a+b\right).

$$\frac{3a-3b+2a+2b}{\left(a+b\right)\left(a-b\right)}-\frac{1}{a^{2}-b^{2}}$$

Combine like terms in 3a-3b+2a+2b.

$$\frac{5a-b}{\left(a+b\right)\left(a-b\right)}-\frac{1}{a^{2}-b^{2}}$$

Factor a^{2}-b^{2}.

$$\frac{5a-b}{\left(a+b\right)\left(a-b\right)}-\frac{1}{\left(a+b\right)\left(a-b\right)}$$

Since \frac{5a-b}{\left(a+b\right)\left(a-b\right)} and \frac{1}{\left(a+b\right)\left(a-b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5a-b-1}{\left(a+b\right)\left(a-b\right)}$$

Expand \left(a+b\right)\left(a-b\right).

$$\frac{5a-b-1}{a^{2}-b^{2}}$$

Differentiate w.r.t. a

$$\frac{2a-5b^{2}+2ab-5a^{2}}{\left(a^{2}-b^{2}\right)^{2}}$$