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 and are differentiable and The quotient rule states that the derivative of 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 properties

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

Proof using implicit differentiation

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

Proof using the chain rule

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 formulas

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.
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