Solve 2ab/a^2-b^2-b/a-b+3

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

Evaluate

$$\frac{3a+4b}{a+b}$$

Short Solution Steps

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

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

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

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

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

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

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

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

Combine like terms in 2ab-ba-b^{2}.

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

Factor the expressions that are not already factored in \frac{-b^{2}+ab}{\left(a+b\right)\left(a-b\right)}.

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

Cancel out a-b in both numerator and denominator.

$$\frac{b}{a+b}+3$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{a+b}{a+b}.

$$\frac{b}{a+b}+\frac{3\left(a+b\right)}{a+b}$$

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

$$\frac{b+3\left(a+b\right)}{a+b}$$

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

$$\frac{b+3a+3b}{a+b}$$

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

$$\frac{4b+3a}{a+b}$$

Differentiate w.r.t. a

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