Solve b/a^2-b^2+(1-a/a+b)

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Do the multiplications in b+a^{2}-b^{2}-a\left(a-b\right).

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

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

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

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

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

Differentiate w.r.t. b

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