# Let f \text{ and } g be the functions and consider the information given in the table...

## Question:

Let {eq}f \text{ and } g {/eq} be the functions and consider the information given in the table below.

$$x$$$$f(x)$$$$f'(x)$$$$g(x)$$$$g'(x)$$
$$1$$$$3$$$$2$$$$4$$$$6$$
$$2$$$$1$$$$8$$$$5$$$$7$$
$$3$$$$7$$$$2$$$$7$$$$9$$

(a) If {eq}F(x) = f(f(x)) {/eq}, find {eq}F'(2) {/eq}.

(b) If {eq}G(x) = g(g(x)) {/eq}, find {eq}G'(3) {/eq}.

## Chain Rule:

Recall that the chain rule instructs us to differentiate a function from the outside in, essentially creating a chain of derivatives that need to be multiplied by each other. We often write it

{eq}[f(g(x))]' = f'(g(x)) \ g'(x) {/eq}

For us, note that {eq}g (x) = f (x) {/eq}, so we will need to be careful.

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Part A

We use the chain rule to find

{eq}\begin{align*} F' (x) &= \frac{d}{dx} [ F (x) ] \\ &= \frac{d}{dx} [ f ( f (x) ) ] \\ &= f' ( f (x) )\... Using the Chain Rule to Differentiate Complex Functions

from

Chapter 8 / Lesson 6
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Learn how to differentiate complex functions using the chain rule. Review an explanation of the chain rule and how to use it to solve complex problems like functions without parentheses and trig functions.