Quotient rule

In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions.[1][2][3] Let where both g and h are differentiable and The quotient rule states that the derivative of f(x) is


  1. A basic example:
  2. The quotient rule can be used to find the derivative of   as follows.


Proof from derivative definition and limit propertiesEdit

Let   Applying the definition of the derivative and properties of limits gives the following proof.


Proof using implicit differentiationEdit

Let   so   The product rule then gives   Solving for   and substituting back for   gives:


Proof using the chain ruleEdit

Let   Then the product rule gives


To evaluate the derivative in the second term, apply the power rule along with the chain rule:


Finally, rewrite as fractions and combine terms to get


Higher order formulasEdit

Implicit differentiation can be used to compute the nth derivative of a quotient (partially in terms of its first n − 1 derivatives). For example, differentiating   twice (resulting in  ) and then solving for   yields



  1. ^ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 0-495-01166-5.
  2. ^ Larson, Ron; Edwards, Bruce H. (2009). Calculus (9th ed.). Brooks/Cole. ISBN 0-547-16702-4.
  3. ^ Thomas, George B.; Weir, Maurice D.; Hass, Joel (2010). Thomas' Calculus: Early Transcendentals (12th ed.). Addison-Wesley. ISBN 0-321-58876-2.