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### Course: AP®︎/College Calculus BC > Unit 3

Lesson 2: The chain rule: further practice- Worked example: Chain rule with table
- Chain rule with tables
- Derivative of aˣ (for any positive base a)
- Derivative of logₐx (for any positive base a≠1)
- Derivatives of aˣ and logₐx
- Worked example: Derivative of 7^(x²-x) using the chain rule
- Worked example: Derivative of log₄(x²+x) using the chain rule
- Worked example: Derivative of sec(3π/2-x) using the chain rule
- Worked example: Derivative of ∜(x³+4x²+7) using the chain rule
- Chain rule capstone
- Proving the chain rule
- Derivative rules review

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# Worked example: Chain rule with table

Through a worked example, we explore the Chain rule with a table. Using specific x-values for functions f and g, and their derivatives, we collaboratively evaluate the derivative of a composite function F(x) = f(g(x)). By applying the chain rule, we illuminate the process, making it easy to understand.

## Want to join the conversation?

- is there anybody who can prove this chain rule?(4 votes)
- Sal has a great explanation for it later in the course:

https://en.khanacademy.org/math/differential-calculus/dc-chain/dc-more-chain-rule/a/proving-the-chain-rule(5 votes)

- Answer is 40, cuz f'(-2)=5 ... isnt?(2 votes)
- No, we are trying to use the Chain Rule here.

d/dx f(g(x)) = f'(g(x))g'(x)

when x = 4, g(4) = -2

when x = -2, f'(-2) = 1

when x = 4, g'(4) = 8

1 * 8 = 8

QED(5 votes)

- What does f-prime and g-prime mean?(1 vote)
- f', read as ef-prime, is the derivative of the function f. Likewise for g.(6 votes)

- Given the chart, how would you know which part (top or bottom) applies to f and g?(3 votes)
- It is explained in the video. You may need to watch it again.(2 votes)

- I need a proof for this magical rule(2 votes)
- Why do you have to multiply by g(x) in the chain rule?

Why is it not just f'(g'(x))?(2 votes)- Are you aware of Leibnitz notation?(You know, the d/dx thing)

An intuition of the chain rule is that for an f(g(x)),

df/dx =df/dg * dg/dx.

If you look at this carefully, this is the chain rule.(2 votes)

- find the equation of the tangent line of f(x) at x=4.(0 votes)
- estimate the limit of 𝑎x−1/ℎ as ℎ→0 using technology, for various values of 𝑎>0(0 votes)

## Video transcript

- [Voiceover] The following
table lists the values of functions f and g and of their derivatives,
f-prime and g-prime for the x values negative two and four. And so you can see for
x equals negative two, x equals four, they gave us the values of f, g, f-prime, and g-prime. Let function capital-F be
defined as the composition of f and g. It's lowercase-f of g of x, and they want us to
evaluate f-prime of four. So you might immediately
recognize that if I have a function that can be
viewed as the composition of other functions that the
chain rule will apply here. And so, and I'm just gonna
restate the chain rule, the derivative of capital-F is going to be the
derivative of lowercase-f, the outside function with
respect to the inside function. So lowercase-F-prime of g of x times the derivative
of the inside function with respect to x times g-prime of x. And if we're looking for F-prime of four, F-prime of four, well everywhere we see an x
we replace it with a four. That's gonna be
lowercase-f-prime of g of four times g-prime of four. Now how do we figure this out? They haven't given us
explicitly the values of the functions for all xs, but they've given it to us
at some interesting points. So the first thing you
might wanna figure out is well what is g of four going to be? Well they tell us: when x is equal to four, g of four is negative two. This tells us that the
value of g of x takes on when x is equal to four is negative two. So this right over here is negative two. And so this first part is
f-prime of negative two. So what is f-prime, what is f-prime of negative two? Well when x is equal to negative two, f-prime is equal to one. So this right over here is
f-prime of negative two. That is equal to one. And now we just have to figure
out what g-prime of four is. Well when, let me circle this, g-prime of four, when x is equal to four, and
I'll scroll down a little bit, when x is equal to four, g-prime
takes on the value eight. So there you have it. F-prime of four is
equal to one times eight which is equal to eight, and we're done.