Solve 1/1-r-r/1-r^2

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

Evaluate

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

Short Solution Steps

Factor 1-r^{2}.

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

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

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

Since \frac{-\left(r+1\right)}{\left(r-1\right)\left(r+1\right)} and \frac{-r}{\left(r-1\right)\left(r+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -\left(r+1\right)-\left(-r\right).

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

Combine like terms in -r-1+r.

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

Expand \left(r-1\right)\left(r+1\right).

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

Differentiate w.r.t. r

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