Solve 1/x+1-1/x+3

$$\frac { 1 } { x + 1 } - \frac { 1 } { x + 3 }$$

Evaluate

$$\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and x+3 is \left(x+1\right)\left(x+3\right). Multiply \frac{1}{x+1} times \frac{x+3}{x+3}. Multiply \frac{1}{x+3} times \frac{x+1}{x+1}.

$$\frac{x+3}{\left(x+1\right)\left(x+3\right)}-\frac{x+1}{\left(x+1\right)\left(x+3\right)}$$

Since \frac{x+3}{\left(x+1\right)\left(x+3\right)} and \frac{x+1}{\left(x+1\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x+3-\left(x+1\right)}{\left(x+1\right)\left(x+3\right)}$$

Do the multiplications in x+3-\left(x+1\right).

$$\frac{x+3-x-1}{\left(x+1\right)\left(x+3\right)}$$

Combine like terms in x+3-x-1.

$$\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Expand \left(x+1\right)\left(x+3\right).

$$\frac{2}{x^{2}+4x+3}$$

Differentiate w.r.t. x

$$\frac{4\left(-x-2\right)}{\left(\left(x+1\right)\left(x+3\right)\right)^{2}}$$