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)

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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)

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do you have any idea how I got the formula u'(x)=f'(g(x))g'(x) in the last problem?
no
that is the chain rule I don't see how you can solve these problems without it: \[\large [f(g(x))]'=f'(g(x))g'(x)\]

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Other answers:

we differentiate our way inward when we have nested functions like this -derivative of the outer function times the derivative of the inner function
ok
so see if you can apply that to find the derivative of\[\large F(x)=f(x^3)\]
3x^2
not quite, it's 3x^2f'(x^3) by the chain rule
oh ok
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
3(4)(6)
72
yes!
now try to come up with an expression for G'(x) remember the chain rule
3f(x)^2f'(x)
nice! now you've got those quantities as well, so what's G'(a) ?
24
thanks for all of the help. i really appreciate your patience with me, especially the last problem where i was making alot of mistakes
alot of the people whom i seek help from would just belittle me
it has discourage me from seeking help face to face
you have done most of the work yourself, don't let them get you down you picked that up at lightning speed :D
new things are always tricky, I'm just glad I could help good luck, and don't forget the chain rule!
alright thanks. i am done for the day. take care

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