Well, I would start with the initial equation:
. Incidentally, the equation , if true for all , implies that for all . Not sure that's where you want to go.
Try differentiating both sides of the initial equation. What does that give you?
I am asked to:
Use the chain rule to show that the derivative of an odd function is an even function.
Basically assuming that f(x) is odd:
f(-x) = -f(x) and f(x) = -f(x)
Show that:
f'(x) = f'(-x)
All must be done in terms of f. How would I go about in doing this?
Then chain rule says:
So we have:
and
Thus:
so differentiating with the cain rule gets:
And since:
we can move over the negitive in the original equation you gave to get:
and
thus:
I feel like there is an error in this proof, but it seems logical, atleast I think I'm going in the right direction. Good luck.