First, let’s review composite functions. A composite function is the combination of two functions, usually written as (f o g)(x) or f(g(x)). The best way to understand composite functions is to envision one as a series of two machines:



Do not confuse a composite function with the product of two functions.

An example of a composite function is h(x) = sin(5x), where f(x)=sin(g(x)) and g(x) = 5x. Now, let’s ask ourselves: How would we differentiate this function?

Other composite functions like h(x) = (x+1)5 can be differentiated by expanding first and then applying the power rule, but that is a long-winded process. Luckily, the process becomes shorter when you use binomial expansion, but what about the function, h(x) = (x3 + x2 + x + 1)5? You can’t use binomial expansion, so it becomes extremely tedious. And in the case of h(x) = sin(5x), as well as in the case of h(x) = (ex + ln(x))7, you cannot simply rely simply on rules you know already, like the power rule......

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Title: Chain Rule
Approximate Word Count: 454
Approximate Pages: 2 (250 words per double-spaced page)

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