Solve 1/k-r+4r/k^2-r^2+2/k+r

$$\frac { 1 } { k - r } + \frac { 4 r } { k ^ { 2 } - r ^ { 2 } } + \frac { 2 } { k + r }$$

Evaluate

$$\frac{3}{k-r}$$

Short Solution Steps

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

$$\frac{1}{k-r}+\frac{4r}{\left(r+k\right)\left(-r+k\right)}+\frac{2}{k+r}$$

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

$$\frac{r+k}{\left(r+k\right)\left(-r+k\right)}+\frac{4r}{\left(r+k\right)\left(-r+k\right)}+\frac{2}{k+r}$$

Since \frac{r+k}{\left(r+k\right)\left(-r+k\right)} and \frac{4r}{\left(r+k\right)\left(-r+k\right)} have the same denominator, add them by adding their numerators.

$$\frac{r+k+4r}{\left(r+k\right)\left(-r+k\right)}+\frac{2}{k+r}$$

Combine like terms in r+k+4r.

$$\frac{5r+k}{\left(r+k\right)\left(-r+k\right)}+\frac{2}{k+r}$$

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

$$\frac{5r+k}{\left(r+k\right)\left(-r+k\right)}+\frac{2\left(-r+k\right)}{\left(r+k\right)\left(-r+k\right)}$$

Since \frac{5r+k}{\left(r+k\right)\left(-r+k\right)} and \frac{2\left(-r+k\right)}{\left(r+k\right)\left(-r+k\right)} have the same denominator, add them by adding their numerators.

$$\frac{5r+k+2\left(-r+k\right)}{\left(r+k\right)\left(-r+k\right)}$$

Do the multiplications in 5r+k+2\left(-r+k\right).

$$\frac{5r+k-2r+2k}{\left(r+k\right)\left(-r+k\right)}$$

Combine like terms in 5r+k-2r+2k.

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

Factor the expressions that are not already factored in \frac{3r+3k}{\left(r+k\right)\left(-r+k\right)}.

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

Cancel out r+k in both numerator and denominator.

$$\frac{3}{-r+k}$$

Differentiate w.r.t. k

$$-\frac{3}{\left(k-r\right)^{2}}$$