Solve 1/x+5-1/x+4

$$\frac { 1 } { x + 5 } - \frac { 1 } { x + 4 }$$

Evaluate

$$-\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+5 and x+4 is \left(x+4\right)\left(x+5\right). Multiply \frac{1}{x+5} times \frac{x+4}{x+4}. Multiply \frac{1}{x+4} times \frac{x+5}{x+5}.

$$\frac{x+4}{\left(x+4\right)\left(x+5\right)}-\frac{x+5}{\left(x+4\right)\left(x+5\right)}$$

Since \frac{x+4}{\left(x+4\right)\left(x+5\right)} and \frac{x+5}{\left(x+4\right)\left(x+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x+4-\left(x+5\right)}{\left(x+4\right)\left(x+5\right)}$$

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

$$\frac{x+4-x-5}{\left(x+4\right)\left(x+5\right)}$$

Combine like terms in x+4-x-5.

$$\frac{-1}{\left(x+4\right)\left(x+5\right)}$$

Expand \left(x+4\right)\left(x+5\right).

$$\frac{-1}{x^{2}+9x+20}$$

Differentiate w.r.t. x

$$\frac{2x+9}{\left(\left(x+4\right)\left(x+5\right)\right)^{2}}$$