**Example**

For example, derivative of (x + 1)/(x^2 + x) is calculated as follows:

By quotient rule, D ( (x + 1)/(x^2 + x) ) is equal to

((x^2 + x) * D (x + 1) - (x + 1) * D (x^2 + x) ) / ((x^2 + x)^2)Now derivative of (x + 1) is 1 and derivative of x^2 + x is 2x + 1 (by the sum rule and power rule). So you get

((x^2 + x) * 1 - (x + 1) * (2x + 1) ) / ((x^2 + x)^2)Simplifying it algebraically, we get

-1/x^2

**Applies (and not applies) to**

The quotient rule applies to quotients of

- variables. Examples: x/y, a/b, 3a/5b
- expressions. Examples: (3a + b)/(2a - b), (x^2 + 4x + 5)/(x + y), sin(x)/cos(x)
- functions. Example: f(x) / g(x)

- constants. Examples: 2/3, pi/e
- a constant and a variables/expression/function: 5/(x - 1), (x^2 - 3x)/10, 15/log(x)