This gets a little involved. Define f by
For we can find f'(x) by using the standard rules: it's . This doesn't have a limit as x goes to 0: if you look at the sequence (which goes to 0), , but looking at the sequence , . Therefore does not exist.
However, f'(0) does exist. Use the definition of derivative:
Since , the difference quotient satisfies
for all . But , so
by the pinching theorem. Therefore f'(0) exists and equals 0.
Here's a picture of f: