Calculus of a Single Variable, Chapter 2, 2.1, Section 2.1, Problem 19

By limit process, the derivative of a function f(x) is :-
f'(x) = lim h --> 0 [{f(x+h) - f(x)}/h]
Now, the given function is :-
f(x) = (x^3) - 12x
thus, f'(x) = lim h --> 0 [{{(x+h)^3} - 12(x+h)} - ((x^3) - 12x)}/h]
or, f'x) = lim h --> 0 [{{(x+h)^3} - (x^3) - 12h}/h]
or, f'(x) = lim h --> 0 [{(x^3) + (h^3) + 3x(h^2) + 3(x^2)h - 12h - (x^3)}/h]
or, f'(x) = lim h --> 0 [{(h^3) -12h + 3x(h^2) + 3(x^2)h}/h]
= [(h^2) + 3xh + 3(x^2) - 12]
putting the value of h = 0 in the above expression we get
f'(x) = 3(x^2) - 12