Calculus: The Quotient Rule

Why it's important

You can use the quotient rule to answer questions like:
Find f'(x) when f(x) = (3 + x2)/(x4 + x). 

What it is

I recite this rhyme to remember the quotient rule:
Low Dee High minus High Dee Low
Draw the Bar and Square Below
Which means:
f'(x) = [low * dee high - high * dee low] / low2
Dee high means the derivative of the high function. You can guess which that is.
In our example, low = x4 + x and high = 3 + x2, so dee low = 4x3 + 1 and dee high = 2x.
f'(x) = [(x4 + x) * (2x) - (3 + x2) * (4x3 + 1)] / [(x+ x)]2
That's it. That's the quotient rule.


I like applying rules I just learned to cases where I know what the answer will be. This helps me build my confidence that I'm using the rule correctly.
x2 is sort of my go-to function to do this. Rewriting it as x2/1 and applying the quotient rule, one gets
[(1) * (2x) - (x2) * 0] / (1)= 2x/1 = 2x
One can also verify the quotient rule using the product rule:
If f(x) = g(x)/h(x) = g(x)*h(x)-1, then
f'(x) = g'(x)*h(x)-1 + g(x)*-1*h(x)-2*h'(x) = h(x)-2 * [g'(x)*h(x) - g(x)*h(x)] = 
[h(x)*g'(x) - g(x)*h'(x)] / [h(x)2] = result from applying the quotient rule.
And in the example I've been using,
f(x) = (3 + x2)/(x4 + x) = (3 + x2)*(x4 + x)-1
f'(x) = (2x) * (x4 + x)-1 + (3 + x2) * -1 * (x4 + x)-2 * (4x3 + 1) =
(x4 + x)-2 * [ 2x * (x4 + x) - (3 + x2) * (4x3 + 1) ] = 
[(x4 + x) * (2x) - (3 + x2) * (4x3 + 1)] / [(x4 + x)]2

Final Notes

The best way to learn the various rules for taking derivatives is to practice them, moving from the easier rules to the more complex ones. It is much harder to learn the quotient rule if you don't know the power rule. It's impossible to apply the quotient rule accurately when a cosine is involved if you don't know the chain rule. And you can't verify the quotient rule using the product rule if you don't know the product rule.
The quotient rule is actually fairly easy to use. The rhyme will always give you the right answer.
But keeping the various rules straight can be difficult. And on the AP calculus tests, there is not much time to try to recall how the various rules work. It's much better to have them down cold, so you can apply them with confidence.
The way to memorize them is to use the rules in easy cases, over and over again, verify the rules worked as expected, and then go on to using them in more complex cases.
Try using Wolfram Alpha or another application to check your results if you're unsure of your answers. Do practice problems in your book and look up the solutions. Make flash cards. Make up your own problems. Quiz your study partners.
Math is like any other subject. You wouldn't go into your history test without spending hours memorizing key events and their significance. You wouldn't take your chemistry test without learning how to use atomic numbers and masses.
And you should likewise go into your calculus test confident that you know how to apply the rules. The studying will only take a few hours for each classroom test, but a 4 or 5 on the AP exam will save you at least a semester if not a year of work in college. And you may even find that you come to enjoy calculus, vastly expanding your possible courses of study in college and perhaps changing your career direction entirely.


Bryce S.

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