What version of the definition of derivative

[Niaboc67]

If my understanding is correct the definition of a derivative is lim h->0 (f(x+h)-f(x))/h However, I've also seen this used: lim x->c (f(x)-f(x))/(x-c) are these both considered valid definition for the derivative or does the derivative have to tend towards zero? I am a bit confused because I see these two versions alternating and wondering which one I should use and where. And my teacher is picky about definitions and notation so I don't know which one he would like to see on the exam.

Thank you

 

[andrewkirk]

The second one should be lim x->c (f(x)-f(c))/(x-c), not the version you wrote.
Note that the first one defines the derivative at x, ie f'(x), while the second defines the derivative at c, ie f'(c).

Both are valid, since they are equivalent, via the substitutions x<-->c, h<-->x-c. I have seen the first more often than the second. I suggest you use whichever version your teacher taught you.