CHAIN RULE

First, permit’s review involved rights. A intricate escape is the combination of deuce 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 conf hold 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(a) composite functions like h(x) = (x+1)5 can be differentiated by expanding first and indeed applying the power tower, but that is a impractical 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 passing 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 pick out already, like the power rule and the rules for differentiating trigonometric, exponential, and logarithmic functions. You need to use the “ scope rule.



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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 inning h(x) = f′(g(x)) g′(x).

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