Quotient rule

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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 properties[edit]

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

Proof using implicit differentiation[edit]

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

Proof using the chain rule[edit]

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[edit]

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 978-0-495-01166-8.
  2. ^ Larson, Ron; Edwards, Bruce H. (2009). Calculus (9th ed.). Brooks/Cole. ISBN 978-0-547-16702-2.
  3. ^ Thomas, George B.; Weir, Maurice D.; Hass, Joel (2010). Thomas' Calculus: Early Transcendentals (12th ed.). Addison-Wesley. ISBN 978-0-321-58876-0.