Calculus Facts
Learning the Product Rule to Avoid Quotient Rule Mistake

This page is a plea to teachers to teach and students to learn the product rule in such a way that reduces the likelihood of making a common mistake in using the quotient rule.

A common mistake when using the quotient rule is to reverse the terms in the numerator, that is, to think that the quotient rule is:

quotient rule

Some examples of this mistake are shown in the Calculus Mistakes section.

In my experience the students who make this mistake have learned the product rule in the form

quotient rule

It seems that the students who have learned product rule using the pattern that the term f(x)g'(x) is followed by the term f'(x)g(x) then incorrectly extend this order of terms to the quotient rule, for which the order matters, due to the subtraction. This suggests a simple way to avoid mislearning the order of terms in the numerator of the quotient rule - learn the product rule with the terms in the order f'(x)g(x) followed by f(x)g'(x):

quotient rule

(The order of the factors in each product does not matter, so the term g(x)f'(x) followed by the term f(x)g'(x) is just as good.)

When the student naturally transfers the order of the terms from this preferred statement of the product rule to the quotient rule, the correct quotient rule is obtained:

quotient rule

This simple suggestion can eliminate a common mistake made when using the quotient rule.

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