Where is the error in this derivation of the derivative of $f(x)=\ln x$?
$$f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}=\lim_{h\rightarrow0}\frac{\ln(x+h)-\ln x}{h}=\lim_{h\rightarrow0}\frac{\ln\left(\frac{x+h}{x}\right)}{h}=\lim_{h\rightarrow0}\frac{\ln\left(1+\frac{h}{x}\right)}{h}$$
$$=\lim_{h\rightarrow0}\ln\left(1+\frac{h}{x}\right)^{1/h}=?$$
Everything in the proof is correct so far. Final step:
$\lim_{h\rightarrow0}\ln\left(1+\frac{h}{x}\right)^{1/h} = \lim_{y \to \infty} \ln ((1+\frac{1}{xy})^y = \ln(e^{\frac{1}{x}}) = \frac{1}{x}$