Solve b^2/a+bdiv(b/a-b-ab/a+b)

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

Evaluate

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

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{b}{a-b} times \frac{a+b}{a+b}. Multiply \frac{ab}{a+b} times \frac{a-b}{a-b}.

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

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

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

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

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

Divide \frac{b^{2}}{a+b} by \frac{ba+b^{2}-a^{2}b+ab^{2}}{\left(a+b\right)\left(a-b\right)} by multiplying \frac{b^{2}}{a+b} by the reciprocal of \frac{ba+b^{2}-a^{2}b+ab^{2}}{\left(a+b\right)\left(a-b\right)}.

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

Cancel out a+b in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out b in both numerator and denominator.

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

Expand the expression.

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

Differentiate w.r.t. b

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