Use the quotient rule to find derivatives of quotients of differentiable functions
Simplify the differentiation process by choosing the correct derivative rules
I can apply the quotient rule appropriately
I can choose the best rule for finding the derivative of a complex function
Quick Lesson Plan
We meet the last “procedural” derivative rule of Unit 2 today. While the derivatives of sin x, or cos x, or ln x are memorized, the Quotient Rule provides students with a process (as did the Product Rule). They will encounter many different functions in the numerators and denominators of “quotient functions,” so remembering the process for dealing with those functions is really important. And really confusing. This lesson will help a lot.
Right next to the chain rule, this is one of the most commonly mis-applied derivative rules out there. To impress upon students the complexity of the Quotient Rule, today we will let them make a conjecture, test it on their calculators, and then realize they are wrong: we are going to “break” the tool they try to use, thus creating the need for the formula. This topic presents a great opportunity to continue a discussion about the domain of the denominator.
The internet is full of tricks and gimmicks and rhymes for remembering the Quotient Rule (low-de-high high-de-low, or “Bottoms up!”). If you have found success with a particular mnemonic or method, continue using that strategy. The best strategy for any student is the one that works best for them.
The Quotient Rule is present throughout all sections of the exam. Non-calculator derivatives are typically less complex than those that require numerical derivatives.
Beware! Students commonly reverse the ‘order of operations’ for this derivative and first multiply the numerator by the derivative of the denominator. Perfect practice makes perfect, so expect and require correct work on these problems. Adding terms in the numerator is a common mistake. Forgetting to square the denominator is also common. Some teachers have a template for students to complete for the first few assignments and then revert to using the template when student accuracy slides.