Solve 2r/r^2-s^2+1/r+s-1/r-s

$$\frac { 2 r } { r ^ { 2 } - s ^ { 2 } } + \frac { 1 } { r + s } - \frac { 1 } { r - s }$$

Evaluate

$$\frac{2}{r+s}$$

Short Solution Steps

Factor r^{2}-s^{2}.

$$\frac{2r}{\left(r+s\right)\left(r-s\right)}+\frac{1}{r+s}-\frac{1}{r-s}$$

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

$$\frac{2r}{\left(r+s\right)\left(r-s\right)}+\frac{r-s}{\left(r+s\right)\left(r-s\right)}-\frac{1}{r-s}$$

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

$$\frac{2r+r-s}{\left(r+s\right)\left(r-s\right)}-\frac{1}{r-s}$$

Combine like terms in 2r+r-s.

$$\frac{3r-s}{\left(r+s\right)\left(r-s\right)}-\frac{1}{r-s}$$

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

$$\frac{3r-s}{\left(r+s\right)\left(r-s\right)}-\frac{r+s}{\left(r+s\right)\left(r-s\right)}$$

Since \frac{3r-s}{\left(r+s\right)\left(r-s\right)} and \frac{r+s}{\left(r+s\right)\left(r-s\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3r-s-\left(r+s\right)}{\left(r+s\right)\left(r-s\right)}$$

Do the multiplications in 3r-s-\left(r+s\right).

$$\frac{3r-s-r-s}{\left(r+s\right)\left(r-s\right)}$$

Combine like terms in 3r-s-r-s.

$$\frac{2r-2s}{\left(r+s\right)\left(r-s\right)}$$

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

$$\frac{2\left(r-s\right)}{\left(r+s\right)\left(r-s\right)}$$

Cancel out r-s in both numerator and denominator.

$$\frac{2}{r+s}$$

Differentiate w.r.t. r

$$-\frac{2}{\left(r+s\right)^{2}}$$