Quotient rule

Summary

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 f and g are differentiable and The quotient rule states that the derivative of h(x) is

It is provable in many ways by using other derivative rules.

Examples

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Example 1: Basic example

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Given  , let  , then using the quotient rule:

 

Example 2: Derivative of tangent function

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The quotient rule can be used to find the derivative of   as follows:

 

Reciprocal rule

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The reciprocal rule is a special case of the quotient rule in which the numerator  . Applying the quotient rule gives

 

Utilizing the chain rule yields the same result.

Proofs

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Proof from derivative definition and limit properties

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Let   Applying the definition of the derivative and properties of limits gives the following proof, with the term   added and subtracted to allow splitting and factoring in subsequent steps without affecting the value:

 
The limit evaluation   is justified by the differentiability of  , implying continuity, which can be expressed as  .

Proof using implicit differentiation

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Let   so that  

The product rule then gives  

Solving for   and substituting back for   gives:

 

Proof using the reciprocal rule or chain rule

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Let  

Then the product rule gives  

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

 

Substituting the result into the expression gives

 

Proof by logarithmic differentiation

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Let   Taking the absolute value and natural logarithm of both sides of the equation gives

 

Applying properties of the absolute value and logarithms,

 

Taking the logarithmic derivative of both sides,

 

Solving for   and substituting back   for   gives:

 

Taking the absolute value of the functions is necessary for the logarithmic differentiation of functions that may have negative values, as logarithms are only real-valued for positive arguments. This works because  , which justifies taking the absolute value of the functions for logarithmic differentiation.

Higher order derivatives

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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

 

See also

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References

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  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.