Experience First

The quotient rule tends to be one of the most procedural things taught in Calculus. We wanted students to do more than just plug and chug into a formula. Though the quotient rule is much harder to visualize than the product rule, the function f(x)=sin(x)/x actually lends itself very well to seeing the components of the original function in the derivative.

 

The experience starts by students describing the slopes of the function sin(x)/x and noticing that this function is not differentiable at zero since the function is not defined there (and continuity is required for differentiability!) We purposefully have students find the derivative of the numerator and denominator separately as this is the most common mistake students make when finding the derivative of a quotient. In question 5, we ask students to consider if this is a reasonable way to find the derivative. There are many reasons students could give for why it’s not. Since cos x/1 is just the cosine, it is unlikely that this is the derivative since the sine function already has cosine as its derivative and sin(x)/x clearly has different slopes than sin x. Furthermore, cos(x) is always defined but the derivative of sin(x)/x is not. Finally, cos(x) fluctuates periodically between -1 and 1 whereas sin(x)/x has steeper slopes near 0 that then flattens as x increases or gets very small. We hope this line of reasoning will help students think twice before making this common mistake!

 

Students find the true derivative by applying the product rule and then simplifying (hopefully down to just one fraction--if groups do not do this on their own, prod them to simplify further until they have just one fraction). We ask students to make sense of this new derivative equation and why it is (at the very least) more likely to be the derivative. Our students had no problem accepting this as the true derivative since it came from their correct use of the product rule.

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The final step is to help students see patterns between the original function and the derivative. Just like the last lesson, there are limitations to the method of turning any quotient into a product (at least without the chain rule!). The quotient rule is ultimately a shortcut or strategy for finding the derivative of a complicated quotient function simply by looking at the numerator and denominator functions and their derivatives separately.

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Formalize Later

When you debrief the various components of the quotient rule, have students point out similarities and differences to the product rule. Furthermore, students should notice that the order of the terms in the numerator matters! There are several ways of remembering the quotient rule. The chant “low-D-high minus high-D-low all over low-low” (the D refers to derivative, the low is the denominator and the high is the numerator) is commonly used. “Bottoms up” is another method to help students remember that the (unchanged) denominator is the first part of the numerator expression. This simple reminder is sufficient for many students, as this tends to be the most common mistake, and the rest follows a predictable pattern.