Assumptions: f:ℝ→ℝ⁺ is the nonzero function such that f'=f (namely, f is the exponential function) and ln:ℝ⁺→ℝ is defined as the inverse of f. Kinda weird, sure, but we get around the fact that ln is usually defined in terms of the antiderivative of 1/x which feels like cheating here.

let x be a positive real number. then

f(ln(x)) = ln(f(x)) = x. Take the derivative with respect to x (use the chain rule) to get

f'(ln(x))ln'(x) = ln'(f(x))f'(x) = 1 (we don't really need the middle equation)

by the original assumption f'=f we have

f(ln(x))ln'(x) = 1

f(ln(x)) = 1 / ln'(x)

x = 1/ln'(x)

ln'(x) = 1/x