"Magical" identity between primitives.

Question

Well, we mathematicians know that (given some "usual" hypotheses) $f'(x)=g'(x)\implies f(x)-g(x)=C$, where $C$ is a constant.

My question is about an elementary example in this context which should be surprising. $\arcsin$ and the like is too difficult for my pupils. Is there anything simpler? The point is to show that, given two functions, if their derivative is the same, these functions are necessarily the same, up to a constant.

Answer 1

Try $\ln(x)$ and $\ln(2x)$?

There's not much to this except contriving a pair of formulas which look deceptively different.

Answer 2

Maybe something like this?

(a) Derive a formula for the derivative of $h(x)=f(x)^{g(x)}$.

(b) Use the formula to compute the derivative of $h(x)=x^{\frac{1}{\ln x}}$.

(c) Explain the result.