Lesson 2.3: Differentiating Combinations of Functions
By this point, we’ve covered the differentiation of a few basic functions: monomials and the two elementary trig functions. Obviously, this is woefully inadequate for the majority of functions. There are four “simple” ways to combine two functions and :
, the sum
, the difference
, the product
, the quotient
The functions and can be any functions (though we’ll limit ourselves to constant, monomial, trigonometric, exponential, and logarithmic) of one variable. Additionally, we’ve already dealt with showing that
for a constant ; this is called the Constant Multiple Rule.
We’ll first deal with the sum function . Consider and . It’s clear not only that and , but also that and thus
The astute ones among you will notice that . Coincidence? No. It’s a trivial proof (and a good one for you to do as an exercise) using difference quotients that
and this is known as the Sum Rule.
The rule for the difference function is very similar, and naturally called the Difference Rule:
and can be proven using the proof of the Sum Rule in conjunction with the Constant Multiple Rule with .
If you’re as smart as Leibniz was—and that’s quite smart indeed—you’ll be thinking that . And if you think so, you’re just as wrong as he was. He did fall into that trap briefly, but then saw what’s wrong with that picture: if that’s true, then the derivative of anything can be shown to be zero. Consider , which we can just as easily write as
Now we’ve really got two functions multipled together: and . Therefore and , giving
but we already know that . There’s your disproof by contradiction.
It’s not a particularly easy derivation (though not that horrible) to find the Product Rule for
I’m not aware of any intuitive way to think of it, but the rule is
and the proof is at the link just above. You’ll definitely need to memorize that, but no worries about the order because it’s a sum.
Similarly, it’s incorrect to think that . (Convince yourself that this would imply that the derivative of anything is undefined.) It’s an even uglier derivation than the Product Rule’s, but the Quotient Rule states that
It’s annoying to have to memorize the Quotient Rule, but you certainly need to as it’s frequently useful and a bitch to rediscover every time. Fortunately, generations of students have developed a totally uninspired mnemonic: low dee-high minus high dee-low; draw a line and square the low. “Low” refers to the bottom fraction, , “high” to , “dee-” to differentiating, and the last line refers to making it a fraction. It’s a pretty dumb mnemonic, but I can’t think of any better one. (If you can, you get an A for the day. Seriously, though, let me know).
For your convenience, here’s a summary:
- the Constant Multiple Rule:

- the Sum Rule:

- the Difference Rule:

- the Product Rule:

- the Quotient Rule:

Now thanks to the Quotient Rule—note that this is the only time you’ll ever be thankful for it—we can find derivatives for the other trigonometric functions:
I’d advise against memorizing the derivatives of the cosecant, secant, and cotangent functions; just memorize those for sine, cosine, and tangent and derive the others on the rare occasion you need them.
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