'Proof' that $f''(x)=\frac{f'(x)}{x}$

Question

Consider the following:

$$f''(x)=\lim_{h\to0}\frac{f'(x+h)-f'(x)}{h}$$ Now using L'Hopital's rule (as this is a case of an inderterminate) we have $$f''(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{hx}$$ But this is $$\frac{1}{x}\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$ which is $\frac{1}{x}f'(x)$.

So it would seem that $$f''(x)=\frac{f'(x)}{x}$$ which is quite obviously false. Where is my error?

Thanks in advance.

Answer 1

L'Hôpital's rule actually implies$$\lim_{h\to0}\frac{f(x+h)-f(x)}{hx}=\lim_{h\to0}\frac{f^\prime(x+h)}{x},$$not$$\lim_{h\to0}\frac{f(x+h)-f(x)}{hx}=\lim_{h\to0}\frac{f^\prime(x+h)-f^\prime(x)}{h}.$$As @Brainstorming notes, your derivative was with respect to the wrong variable.