Solve 1/a+b+ab/b^2-a^2

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

Evaluate

$$\frac{ab-a+b}{b^{2}-a^{2}}$$

Short Solution Steps

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

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

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

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

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

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

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

$$\frac{-a+b+ab}{-a^{2}+b^{2}}$$

Differentiate w.r.t. b

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