Quotient rule

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In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist.[1][2][3]

If the function one wishes to differentiate, , can be written as

and , then the rule states that the derivative of is

Many people remember the Quotient Rule by the rhyme "Low D-high, High D-low, cross the line and square the low." It is important to remember the 'D' describes the succeeding portion of the original fraction.

Proof[edit]

First we have our function:

We rewrite the fraction using a negative exponent.

Take the derivative of both sides, and apply the product rule to the right side.

To evaluate the derivative in the second term, apply the chain rule, where the outer function is , and the inner function is .

Rewrite things in fraction form.

Finally, use the least common denominator (which is ) to combine the fractions.

Implicit differentiation and higher order formulas[edit]

The formula can also be derived using implicit differentiation and the product rule. This results in a simpler formula, especially for higher order derivatives. can be implicitly defined as Using the product rule to differentiate this yields Solving for yields

It is much easier to derive higher order quotient rules using implicit differentiation. For example, two implicit differentiations of yields and solving for yields

References[edit]

  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.