First recall the definition of "derivative" for any function f :

f ' (x) = h0 f(x+h) - f(x) (x+h) - x

Now we interpret the functional notation for our particular function f(x) = x³ ;
at this point, you may want to review the binomial theorem :

f ' (x) =

h0

(x+h)3 - x3 (x+h) - x

Now we must rewrite the fraction, since both top and bottom approach zero :

=

h0

[x3 + 3x2h + 3xh2 + h3] - x3 (x+h) - x

Now we cancel two pairs of numbers, colored red above, and factor the top :

=

h0

3x2h + 3xh2 + h3 h

=

h0

h[3x2 + 3xh + h2] h

At this point, both top and bottom are ZERO if h = 0. However, if h is NOT ZERO, we cancel the two red h above:

=

h0

[3x2 + 3xh + h2]

And now the limit can be readily seen: the last two terms shrink to ZERO while the first term remains fixed.