# Help with second derivative

## Homework Statement

if d/dx(f(x)) = g(x) and d/dx(g(x)) = f(x^2), then d^2/dx^2 ( f(x^3) ) = ?

## The Attempt at a Solution

from 1 and 2 we get d.dx(g(x)) = d2/dx2(f(x)) = f(x^2)
but then what? That doesn't tell me anything about f(x^3)

Related Calculus and Beyond Homework Help News on Phys.org
Mark44
Mentor

## Homework Statement

if d/dx(f(x)) = g(x) and d/dx(g(x)) = f(x^2), then d^2/dx^2 ( f(x^3) ) = ?

## The Attempt at a Solution

from 1 and 2 we get d.dx(g(x)) = d2/dx2(f(x)) = f(x^2)
but then what? That doesn't tell me anything about f(x^3)

What would you do to calculate d^2/dx^2 ( f(x) )?

Now, think chain rule.

tiny-tim
Homework Helper
hi shar_p! (try using the X2 icon just above the Reply box )

let's rewrite the question:

f' = g

so what is (f(x3))' ?

Thanks for the hints:
f' = g

so (f(x3))' would be 3 x2 g(x3)

But we need d2/dx2(f(x3)) which is ( 3 x2 g(x3) )'
= 6x.g(x3) + 3x2. (g(x3))'

How do I simplify it further? what is (g(x3))' ?

I guess since this is a multiple choice question and 1 of the answers is 9x4f(x6) + 6x g(x3), I would just choose that one. But I would still like to know how to get 9x4f(x6) from 3x2. (g(x3))'

SammyS
Staff Emeritus
Homework Helper
Gold Member
No, f '(x3) = 3 x2 g'(x3).

tiny-tim
Homework Helper
hi shar_p! But we need d2/dx2(f(x3)) which is ( 3 x2 g(x3) )'
= 6x.g(x3) + 3x2. (g(x3))'

How do I simplify it further? what is (g(x3))' ?

I guess since this is a multiple choice question and 1 of the answers is 9x4f(x6) + 6x g(x3), I would just choose that one. But I would still like to know how to get 9x4f(x6) from 3x2. (g(x3))'
well, you know g'(x) = f(x2),

so g'(x3) = … ?

and so (g(x3))' = … ? No, f '(x3) = 3 x2 g'(x3).
(why the large font? )

nooo, f' = g so f '(x3) = g(x3)

and f( (x3))' = 3 x2 g(x3), as shar_p says

g'(x) = f(x2),

so g'(x3) = f((x3))2 = f(x6)

and so (g(x3))' = … ?

ans = 6x.g(x3) + 3x2. (g(x3))'
= 6x.g(x3) + 3x2.f(x6).3x2
= 6x.g(x3) + 9x4.f(x6)
yes!!

Thanks a lot.

I tried to explain this to my friend and he said that :
if d/dx(f(x)) = g(x) and d/dx(g(x)) = f(x2), then d2/dx2 ( f(x3) ) = ?
since d/dx(g(x)) = d2/dx2(f(x)) = f(x2) ,
d2/dx2(f(x3)) = f((x3)2)=f(x6)

so my initial step of (f(x3))' would be 3 x2 g(x3) is wrong?
since f' = g
f'(x3) = g(x3)

Why is d2/dx2 ( f(x3) ) the same as {f(x3)}'' and not f''(x3)

A. f(x6)
B. g(x3)
C. 3x2. g(x3)
D. 6x.g(x3) + 9x4.f(x6)
E. f(x6) + g(x3)

Last edited:
tiny-tim
Homework Helper
hi shar_p! (just got up :zzz: …)
since d/dx(g(x)) = d2/dx2(f(x)) = f(x2) ,
d2/dx2(f(x3)) = f((x3)2)=f(x6)
sorry, your friend is talking rubbish the last line should be

d2/d(x3)2(f(x3)) = f((x3)2)=f(x6) But I still don't understand why we have to use {f(x3)}'' and not f''(x3).
since f' = g
f'(x3) = g(x3)
f''(x3) = 3x2 g'(x3) = 3x2f(x6)

Last edited:
tiny-tim
Homework Helper
But I still don't understand why we have to use {f(x3)}'' and not f''(x3).
they mean different things …

in the first, the '' is wrt x

in the second, the '' is wrt x3 (suppose f(x) = x2, then f'(x) = 2x and f''(x) = 2 …

so {f(x3)}'' = (x6)'' but f''(x3) = 2)

Ok I think I get it. I need to go from f(x) to f'(x3) first before substituting g and that is what is confusing my friend.