can someone help me solve this problem ?

let f(x)= (x^3+3x+8)^2

F'(x)=
F'(2)=

Question

can someone help me solve this problem ? 

let f(x)= (x^3+3x+8)^2

F'(x)=
F'(2)=

hartnn · Answer

do you know the chain rule ?

anonymous · Answer

not really sure with the chain rule..

anonymous · Answer

need to learn that then to do a question of this type

anonymous · Answer

yes, that's why I'm trying to learn it here.. if that helps.

hartnn · Answer

let me give you a similar example, if 
$f(x) = (ax^2+bx+c)^3 $
then by chain rule, $f'(x)=3(ax^2+bx+c)^2 (ax^2+bx+c)' \  =3(ax^2+bx+c)^2(2ax+b) $
got this ?

hartnn · Answer

by definition, 
$f(g(x))' = f'(g(x)) \times g'(x)$

anonymous · Answer

yeah

hartnn · Answer

so try once for your Q, maybe.. ?

anonymous · Answer

you dont subtract the exponent n-1?

hartnn · Answer

i made the exponent n-1
in the Q, ax^2+bx+c had exponent 3, in the answer i became 2, right ?

hartnn · Answer

for x^n, the derivative is 
$(x^n)'=n x^{n-1}$

anonymous · Answer

okay... sort of there. but then the second equation next to it confused me a little.

hartnn · Answer

yeah, from the definition, i needed to multiply by the derivative of inner function also (g'(x))
so ,i multiplied by derivative of ax^2+bx+c which is 2ax+b

hartnn · Answer

in your Q, you'll multiply by (x^3+3x+8)

hartnn · Answer

**by the derivative of (x^3+3x+8)

anonymous · Answer

so the derivative for F'(x) would be.. (3x^2+3x+8)?

hartnn · Answer

no... you have to first derivate the out function, (x^2), then multiply the derivative of each term of x^3+3x+8
first tell whats the derivative of x^2

anonymous · Answer

2x

hartnn · Answer

so, derivative of (x^3+3x+8)^2 will be  2(x^3+3x+8) *(derivative of x^3+3x+8 ) , right ?
can you differentiate, x^3+3x+8 ?

anonymous · Answer

2x^2+6x+16?

hartnn · Answer

no ? how u got that ?

anonymous · Answer

multiplied...

hartnn · Answer

derivative of  x^3+3x+8
lets take it term by term
derivative of x^3 =... ?

anonymous · Answer

3x

hartnn · Answer

no...
derivative of x^n = nx^{n-1}

anonymous · Answer

3x^2

hartnn · Answer

derivative of 3x ?

anonymous · Answer

3

hartnn · Answer

derivative of 8 ?

anonymous · Answer

2? i really am so confused

hartnn · Answer

derivative of a constant = 0 
so derivative of 8 is 0
so, derivative of x^3+3x+8 = 3x^2+3
right ?

anonymous · Answer

oh okay.

anonymous · Answer

i understand that better.

anonymous · Answer

now what if 2 was plugged in

hartnn · Answer

so, we have :
 derivative of (x^3+3x+8)^2 will be  2(x^3+3x+8) *(derivative of x^3+3x+8 ) , 
= 2(x^3+3x+8) * (3x^2+3)
got this ?

anonymous · Answer

i thought the derivative of f '(x) is 3x^2+3

hartnn · Answer

didn't i mention to u the chain rule ?

anonymous · Answer

oh yeah, multiple use the one before and after

hartnn · Answer

so any doubts now ? that would be your final answer...

anonymous · Answer

nope no doubts. now what if the problem was f ' (2)= ?

hartnn · Answer

just plug in x=2

anonymous · Answer

i did but im still not sure what am i suppose to do.

anonymous · Answer

i plug them in but i get confused with the derivative and actually multiplying it out.

hartnn · Answer

2(x^3+3x+8) * (3x^2+3)
put x=2
2(2^3+3*2+8) * (3*2^2+3) =... ?

anonymous · Answer

44

hartnn · Answer

i get 660 :P using google https://www.google.co.in/search?q=2(2%5E3%2B3*2%2B8)+*+(3*2%5E2%2B3)&aq=f&oq=2(2%5E3%2B3*2%2B8)+*+(3*2%5E2%2B3)&aqs=chrome.0.57.192&sourceid=chrome&ie=UTF-8

anonymous · Answer

thank you. thats correct

hartnn · Answer

welcome ^_^

anonymous · Answer

would it be possible to know the answer of a few problems because I would need to sign off and practice by hand but I would like to check my work and see if they're correct.