How to Use and Remember the Quotient Rule

Calculus, Memorization Tricks

When you are trying to remember anything in math, it helps to notice the similarities and patterns that seem to pop up everywhere (hint: its not a coincidence that this happens). Ever since I first learned calculus, I used a similarity between the quotient rule and the product rule to help myself remember both. First, here is how I think about the product rule if we were finding the derivative of fg:

Write the product out twice adding the two terms
fg+fg
Prime on the first term, prime on the second term
f'g+fg'

If instead we were finding the derivative of \dfrac{f}{g} I would first see if I could simplify and if not, I would go for the quotient rule since this is a quotient. With that said, pay attention to the similarity between this and the product rule:

Write the product out twice SUBTRACTING the two terms
fg-fg
Note: order matters here: numerator*denominator is the “product” I refer to.
Prime on the first term, prime on the second term
f'g-fg'
Divide by the denominator squared
\dfrac{f'g-fg'}{g^2}

Do you see it? In both cases, I am writing the product out twice. Sure, I have more to do with the quotient rule after that but if you think about it, that seems to happen a lot when dealing with functions involving a quotient (yes a very informal way of thinking about this – but again – its just a way to help your brain remember). After that, like the power rule – its just a formula! All the possible “work” will happen as we simplify.

For example, lets use this to find the derivative of \dfrac{\ln(x)}{x}. Since the quotient rule can make things more complicated, I always take a look to see if simplifying will help and here, there is nothing to simplify (don’t fall into the trap of thinking that you can cancel an x here! ln(x) is a function. That would be like cancelling the x in \sqrt(x) and saying you get \sqrt()).

Since there is nothing to simplify and this is a quotient, well, I’m going to use the quotient rule!

Write the product out twice SUBTRACTING the two terms
ln(x)x-ln(x)x
Prime on the first term, prime on the second term
(ln(x))'x-ln(x)x'
Divide by the denominator squared
\dfrac{(ln(x))'x-ln(x)x'}{x^2}

\dfrac{(ln(x))'x-ln(x)x'}{x^2}=\dfrac{(\frac{1}{x})x-ln(x)}{x^2}=\dfrac{1-ln(x)}{x^2}. (Since (ln(x))'=\dfrac{1}{x}).

Did I pick an easier example here? Sure, but this is just about the mechanics of the quotient rule. We will look a little deeper in a future post!

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