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 and the rules for differentiating trigonometric, exponential, and logarithmic functions. You need to use the “chain rule.”

The Chain Rule:

If f is differentiable at the point u = g(x), and g is differentiable at x, then the composite function (f o g)(x) = f(g(x)) is differentiable at x, and

(f o g)′(x) = f′ (g(x)) g′(x)

In Leibniz notation, if y = f(u) and u = g(x), then

where dy/du is evaluated at u = g(x).

Now that we know the chain rule, let’s apply it to the function h(x) = sin(5x) and find its derivative at x = 2π/5.

You can use the chain rule repeatedly like in the case of g(t) = tan(5 – sin(2t)). Find its derivative.

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The chain rule now presents us the opportunity to integrate functions with the form h(x) = f′(g(x)) g′(x).

We know that a function is equal to...