Solve 4/a+b+5/a-b-10b/a^2-b^2

$$\frac { 4 } { a + b } + \frac { 5 } { a - b } - \frac { 10 b } { a ^ { 2 } - b ^ { 2 } }$$

Evaluate

$$\frac{9}{a+b}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+b and a-b is \left(a+b\right)\left(a-b\right). Multiply \frac{4}{a+b} times \frac{a-b}{a-b}. Multiply \frac{5}{a-b} times \frac{a+b}{a+b}.

$$\frac{4\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}+\frac{5\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{10b}{a^{2}-b^{2}}$$

Since \frac{4\left(a-b\right)}{\left(a+b\right)\left(a-b\right)} and \frac{5\left(a+b\right)}{\left(a+b\right)\left(a-b\right)} have the same denominator, add them by adding their numerators.

$$\frac{4\left(a-b\right)+5\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{10b}{a^{2}-b^{2}}$$

Do the multiplications in 4\left(a-b\right)+5\left(a+b\right).

$$\frac{4a-4b+5a+5b}{\left(a+b\right)\left(a-b\right)}-\frac{10b}{a^{2}-b^{2}}$$

Combine like terms in 4a-4b+5a+5b.

$$\frac{9a+b}{\left(a+b\right)\left(a-b\right)}-\frac{10b}{a^{2}-b^{2}}$$

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

$$\frac{9a+b}{\left(a+b\right)\left(a-b\right)}-\frac{10b}{\left(a+b\right)\left(a-b\right)}$$

Since \frac{9a+b}{\left(a+b\right)\left(a-b\right)} and \frac{10b}{\left(a+b\right)\left(a-b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{9a+b-10b}{\left(a+b\right)\left(a-b\right)}$$

Combine like terms in 9a+b-10b.

$$\frac{9a-9b}{\left(a+b\right)\left(a-b\right)}$$

Factor the expressions that are not already factored in \frac{9a-9b}{\left(a+b\right)\left(a-b\right)}.

$$\frac{9\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}$$

Cancel out a-b in both numerator and denominator.

$$\frac{9}{a+b}$$

Differentiate w.r.t. a

$$-\frac{9}{\left(a+b\right)^{2}}$$