Solve 1/1-a^{2/b^2}+b/a+b+a^2/a^2-b^2

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

Evaluate

$$\frac{a+2b}{a+b}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{b^{2}}{b^{2}}.

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

Since \frac{b^{2}}{b^{2}} and \frac{a^{2}}{b^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Divide 1 by \frac{b^{2}-a^{2}}{b^{2}} by multiplying 1 by the reciprocal of \frac{b^{2}-a^{2}}{b^{2}}.

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

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

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

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}.

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

Since \frac{b^{2}}{\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, add them by adding their numerators.

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a+b\right)\left(-a+b\right) and \left(a+b\right)\left(a-b\right) is \left(a+b\right)\left(-a+b\right). Multiply \frac{a^{2}}{\left(a+b\right)\left(a-b\right)} times \frac{-1}{-1}.

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

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

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

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

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

Cancel out -a+b in both numerator and denominator.

$$\frac{a+2b}{a+b}$$

Differentiate w.r.t. b

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