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### Quotient rule - Derivatives

The quotient rule helps calculate the derivative of a quotient (that is a rational expression or a division of two terms). For any two terms, variables, expressions, or functions expressed as u/v, their derivative is:
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)
while the quotient rule does not apply quotients of
• constants. Examples: 2/3, pi/e
• a constant and a variables/expression/function: 5/(x - 1), (x^2 - 3x)/10, 15/log(x)
When you have a double quotient, that is, a rational expression over another rational expression, for example ((x + 1)/(x^2 + x)) /((x- 2)/(x^2 + 2x)) then the quotient rule is applied turn by turn on first the whole rational expression, and then individually on each rational expression in the numerator and denominator.