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)
Bg(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 was saying, but then I was confused by what the second half said. I don’t know how to proceed with this question 🙁 (f'(x)=d/dx(f(x)), I use both forms when 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 the product rule to derive.
= 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 bit of rearranging, we can see that the answer is D. It’s important not to forget the chain rule at any step in this question, and to apply the product rule when we had to 🙂 d/dx f(x^3) = df/dx * 3x^2
then,
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 starting to look like D, right? ws wx