Proofs of derivatives

[jessebu]

The problem is:
Suppose that f: R->R is differentiable and never 0.
(a) Prove that log(abs(f))' = f'/f
(b) Suppose that f' = cf for some constant c. Show that f(x) = ke^(cx) for some number k.

I'm not sure how to get started on either of them.

 

[daon]

The first is just using the chain rule. Unless you had to use the limit process?

The second may use the first:

f' = cf <=> f'/f=c <=> ln(abs(f)) = cx + c' <=> abs(f) = e^(cx+c') = e^(cx)*e^c' = k*e^(cx) where k=e^c'.

if f > 0, you may drop the absolute value sign. otherwise, note if f(x) <0 for some x, that says k=e^(c') < 0, which is never true.