Question

If f and g are the functions whose graphs are shown below, let u(x)=f(g(x)) and v(x)=g(f(x)) .

find u'(3) and v'(3)

Answer 1

start with finding expressions for u'(x) and v'(x)

Answer 2

u'(x)=f'(g(x))g'(x)
v'(x)=g'(f(x))f'(x)

Answer 3

yeah i dont understand this. am i just looking at 3 on the graph?

Answer 4

say you want u'(3), you first need g(3) because g(x) is part of the formula you need
what is g(3) according to the graph?

Answer 5

looks like 4

Answer 6

looks like g(3)=10 to me
each square is 2 vertically, and 1 horizontally as far as I can tell

Answer 7

g(3)=8 I mean, sorry

Answer 8

I'm not really sure about the increments on the graph though, it's a bit vague...

Answer 9

8, how so?

Answer 10

It looks to me like each square vertically is 2 (by the marking), but I guess it must be 4 since 8 is not on the graph horizontally
so ok, g(3)=4
what is the value of f'(4) ?
[remember we need f'(g(3))g'(3)]

Answer 11

also remember that f'(4) represents the slope of f(x) at x=4

Answer 12

f(4) is 2

Answer 13

we want f'(4)
that is the slope of f(x) at x=4
what is the slope of the line in the graph f(x) at x=4 ?

Answer 14

when x is 4, f is 2 right?

Answer 15

yes, but that is the value of f(4) not f'(4)
f'(4) is the SLOPE, not the value of the function
do you remember how to find the slope of a straight line from algebra?

Answer 16

ohh

Answer 17

y2-y1/x2-x1?

Answer 18

actually sorry, they are the same in this case, but that is pure luck! lol
notice that the slope (rise over run) is also 2 !

Answer 19

that is what f' means, the slope of f
so similarly what is the last piece of the formula we need ?
u'(3)=f'(g(3))g'(3)
and we still need g'(3)
what is it?

Answer 20

rise over run right? 4/3? no?

Answer 21

at x=3 it looks to me like g has a slope of -1

Answer 22

how did you figure that out?

Answer 23

look at the tail last portion of g(x), the far right portion of the top graph.
it dips downward
it seems to go down one unit for every unit it goes to the right
rise/run=-1/1=-1

Answer 24

here's a sketch of g

Answer 25

ok i see now i see

Answer 26

here's the part from x=2 to x=4

Answer 27

in your graph you can see it goes one over and one down on that region

Answer 28

(I'm not gonna draw the little squares in...)

Answer 29

i see that thanks

Answer 30

so you have all the pieces now
u'(3)=f'(g(3))g'(3)
you know f'(g(3)) and g'(3), so multiply them to find u'(3)

Answer 31

what is g(3) ?

Answer 32

4
slope is -1

Answer 33

right, g(3)=4
and what is f'(g(3)) ?

Answer 34

f is also 4 right?

so that makes 4 x 4?

Answer 35

no, we want f'(g(3))=f'(4)
because g(3)=3, right ?
what is f'(4) ?