Question

Let F(x)=f(x^ 3 ) and G(x)=(f(x)) ^3 . You also know that a ^2 =4,f(a)=2,f ′ (a)=2,f ′ (a ^3 )=6 . Find F'(a) and G'(a)

Answer 1

do you have any idea how I got the formula u'(x)=f'(g(x))g'(x) in the last problem?

Answer 2

no

Answer 3

that is the chain rule
I don't see how you can solve these problems without it:
\[\large [f(g(x))]'=f'(g(x))g'(x)\]

Answer 4

we differentiate our way inward when we have nested functions like this

-derivative of the outer function times the derivative of the inner function

Answer 5

ok

Answer 6

so see if you can apply that to find the derivative of\[\large F(x)=f(x^3)\]

Answer 7

3x^2

Answer 8

not quite, it's
3x^2f'(x^3)
by the chain rule

Answer 9

oh ok

Answer 10

we start with the derivative of the outer function, then the inner one

so F'(a)=3a^2f'(a^3)
and we are given all those quantities in the problem, so plug in and solve

Answer 11

3(4)(6)

Answer 12

72

Answer 13

yes!

Answer 14

now try to come up with an expression for G'(x)
remember the chain rule

Answer 15

3f(x)^2f'(x)

Answer 16

nice!
now you've got those quantities as well, so what's G'(a) ?

Answer 17

24

Answer 18

thanks for all of the help. i really appreciate your patience with me, especially the last problem where i was making alot of mistakes

Answer 19

alot of the people whom i seek help from would just belittle me

Answer 20

it has discourage me from seeking help face to face

Answer 21

you have done most of the work yourself, don't let them get you down
you picked that up at lightning speed :D

Answer 22

new things are always tricky, I'm just glad I could help
good luck, and don't forget the chain rule!

Answer 23

alright thanks. i am done for the day. take care