Question

If d/dx(f(x))=g(x) and d/dx(g(x))=f(x^2) , then d^2/dx^2(f(x^3))
, then d2dx2(f(x3))=

A f(x^6)
B g(x^3)
C 3x^2g(x^3)
D 9x^4f(x^6) + 6xg(x^3)
E f(x^6) + g(x^3)

I understood what the first half of the question said, but then I was confused by what the second half says. I'm confused on how to proceed with this question :(

Answer 1

d/dx f(x^3) = df/dx * 3x^2
so,
d^2/dx^2 f(x^3) = d/dx (d/dx f(x^3))
= d/dx (df/dx * 3x^2)
= d/dx (df/dx)) * 3x^2 + 6x df/dx
It's beginning to look like D, right?

Answer 2

Ws w x

Answer 3

(f’(x)=d/dx(f(x)), I use both forms while solving.)
d/dx(f(x))=g(x) is defined,
so d^2/d^2x(f(x))=g’(x)dx (chain rule).
d/dx(g(x))=f(x^2) is defined.
Now, we look at what we need to find, the second derivative of f(x^3)
d^2/d^2x f(x^3) = d/dx(f’(x^3)). Plug in using what is defined.
=d/dx(g(x^3)3x^2). Now use product rule to derivate.
= g(x^3)6x + g’(x^3)3x^2. Plug in g’(x)=f(x^2)
=g(x^3)6x+f((x^3)^2)(3x^2)(3x^2)
=g(x^3)6x+f(x^6)9x^4
With a little rearranging, we can see the answer is D. It’s important to not forget the chain rule in any step in this question, and to apply the product rule when we had to :)