Question

Use the table below to evaluate d/dx [g[f(2x]] at x=1.

x 1 2 3 4
f(x) 6 1 8 2
f ′(x) 1 3 5 7
g(x) 1 4 4 3
g ′(x) 4 5 5 –4

Answer 1

Use the table below to evaluate d/dx [g[f(2x)] at x=1.

\(\bf ~~x \quad \quad~ 1\quad \quad~ 2\quad ~~~3\quad~~~4\)
`f(x) 6 1 8 2`
f ′(x) 1 3 5 7
`g(x) 1 4 4 3`
g ′(x) 4 5 5 –4

making it more clear

Answer 2

Apply the chain rule to differentiate the g.

Answer 3

wait don't we have to set it up fiirst

Answer 4

you already have the set up

Answer 5

You need to evaluate g'(1) .... no?

Answer 6

Isn't that the meaning of evaluating the derivative (d/dx) of g(x) at x=1??

Answer 7

So do you agree that you need g'(1) ?

Answer 8

uh I guess not sure. Idk i'm just confused on how to start

Answer 9

The derivative of \(\color{#000000 }{ \displaystyle f(x) }\) is \(\color{#0000ff }{ \displaystyle f'(x) }\).


However, if you have a function of \(x\) inside the \(f(x)\), then
The derivative of \(\color{#000000 }{ \displaystyle f(g(x)) }\) is NOT just \(\color{#ff0000}{ \displaystyle f'(g(x)) }\).

Rather, the derivative of \(\color{#000000 }{ \displaystyle f(g(x)) }\) is (YES) \(\color{#0000ff }{ \displaystyle f'(g(x))\cdot g'(x) }\),
and we get this by applying the Chain Rule.

And what happens when you have another function of x
inside the g(x)- (in your case "2x"), you apply the chain
rule once again.

The derivative of \(\color{#000000 }{ \displaystyle f(g(2x)) }\) is NOT \(\color{#ff0000 }{ \displaystyle f'(g(2x))\cdot g'(2x) }\);
Rather, the derivative of \(\color{#000000 }{ \displaystyle f(g(2x)) }\) is \(\color{#0000ff }{ \displaystyle f'(g(2x))\times g'(2x) \times 2 }\)

Answer 10

I might have said in a too lengthy way, but I just want to make sure you understand that the chain rule is applied multiple times.

Answer 11

no no this makes more sense to me. so since x=1 it's just f'(g(2)) x g'(2) x 2

Answer 12

yes, very good

Answer 13

so now i plug in what's in the chart

Answer 14

yes

Answer 15

f'(4)) x 5 x 2
7 x 10
70

Answer 16

Yes, fabulous !!

Answer 17

Thank you so much ;)

Answer 18

Anytime!

Answer 19

It's domo! :D

Answer 20

domo?

Answer 21

domo icon

Answer 22

yup. someone in a domo suit ;D

Answer 23

lol

Answer 24

.... \( ; ) \)