Finding Derivatives of Composite Functions in Calculus

[lexismone]

Homework Statement

Let f and g be the functions in the table below.

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

Homework Equations

If F(x) = f(f(x)), find F '(2).
If G(x) = g(g(x)), find G'(1).

The Attempt at a Solution

i took F(x)=f(f(x)) meaning when f(x)=3, F(x)= 3(3)=9
and
G(x)=g(g(x)) to mean when g(x)=2 G(x)=2(2)=4

so if i am looking at this correctly, can someone help me on where to go from here. please

 

[Office_Shredder]

No, that's not what it means. f(f(x)) means evaluate f(x), then evaluate f(x) at that point. So for example:

f(f(1))=f(3)=2
g(g(1))=g(2)=3

 

[Mark44]

Homework Statement

Let f and g be the functions in the table below.

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

Homework Equations

If F(x) = f(f(x)), find F '(2).
If G(x) = g(g(x)), find G'(1).

The Attempt at a Solution

i took F(x)=f(f(x)) meaning when f(x)=3, F(x)= 3(3)=9

You need to work on understanding function notation better. From the table f(1) = 3, f(2) = 1, and f(3) = 2.

There is no formula for f(x), so it's meaningless to say that f(x) = 3.

This problem is all about understanding the chain rule. You also need to understand the difference between F'(x) and F'(2).

First, find an expression for F'(x).
Next, evaluate F'(x) at x = 2.

The other problem is exactly the same.

and
G(x)=g(g(x)) to mean when g(x)=2 G(x)=2(2)=4

so if i am looking at this correctly, can someone help me on where to go from here. please

 

[lexismone]

F'(x)=f'(f(x))
G'(x)=g'(g(x))
??

 

[Mark44]

F'(x)=f'(f(x))
G'(x)=g'(g(x))
??

That's a start, but you're missing a factor that comes from the chain rule.