Solve 1/x+5-1/x+8

$$\frac { 1 } { x + 5 } - \frac { 1 } { x + 8 }$$

Evaluate

$$\frac{3}{\left(x+5\right)\left(x+8\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+5 and x+8 is \left(x+5\right)\left(x+8\right). Multiply \frac{1}{x+5} times \frac{x+8}{x+8}. Multiply \frac{1}{x+8} times \frac{x+5}{x+5}.

$$\frac{x+8}{\left(x+5\right)\left(x+8\right)}-\frac{x+5}{\left(x+5\right)\left(x+8\right)}$$

Since \frac{x+8}{\left(x+5\right)\left(x+8\right)} and \frac{x+5}{\left(x+5\right)\left(x+8\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x+8-\left(x+5\right)}{\left(x+5\right)\left(x+8\right)}$$

Do the multiplications in x+8-\left(x+5\right).

$$\frac{x+8-x-5}{\left(x+5\right)\left(x+8\right)}$$

Combine like terms in x+8-x-5.

$$\frac{3}{\left(x+5\right)\left(x+8\right)}$$

Expand \left(x+5\right)\left(x+8\right).

$$\frac{3}{x^{2}+13x+40}$$

Differentiate w.r.t. x

$$\frac{3\left(-2x-13\right)}{\left(\left(x+5\right)\left(x+8\right)\right)^{2}}$$