Solve 1/x+b-2b/x^2-b^2+1/x-b

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x-b-2b.

$$\frac{x-3b}{\left(x+b\right)\left(x-b\right)}+\frac{1}{x-b}$$

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

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

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

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

Combine like terms in x-3b+x+b.

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

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

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

Cancel out x-b in both numerator and denominator.

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

Differentiate w.r.t. x

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