As an alternative to the definition of the concept of derivative by using limits, there is also the definition used in a book title Calculus Unlimited.
The value of the derivative $f'(a)$ is $\ge m$ if (but not only if) there is some open interval about $a$ within which $f(x) \left\{ \begin{array}{c} > \\[4pt] < \end{array} \right\} f(a) + m(x-a)$ according as $x \left\{ \begin{array}{c} > \\[4pt] < \end{array} \right\}a$.
But as for using any definition to find $\dfrac{d}{dx} x^3$, one could simply omit that nonsense and give them problems like this:
$$ (fg)'(x) \overset A = \lim_{w\to x}\frac{f(w)g(w)-f(x)g(x)}{w-x} \overset B = \lim_{w\to x}\frac{\overbrace{f(w)g(w) - f(w)g(x)} + \overbrace{f(w)g(x) - f(x)g(x)}}{w-x} $$
$$ \overset C =\lim_{w\to x} \left(f(w)\frac{g(w)- g(x)}{w-x} + g(x)\frac{f(w)-f(x)}{w-x} \right) $$
$$ \overset D =\left( \lim_{w\to x} f(w)\right) \left(\lim_{w\to x} \frac{g(w)-g(x)}{w-x}\right) + \left(\lim_{w\to x} g(x)\right)\left(\lim_{w\to x} \frac{f(w)-f(x)}{w-x}\right) $$
$$ \overset E = f(x)g'(x) + g(x) f'(x). $$
(a) What statement is proved by the argument above?
(b) One of the steps labeled $A$ through $E$ above uses the definition of "derivative" twice? Identify it and explain your choice.
(c) One of the steps uses the definition of "derivative" just once. Identify it and explain your choice.
(d) Two of the steps use only algebra and require no knowledge of limits. Identify them and explain.
(e) One of the steps uses properties of limits discussed in Chapter 2. Identify it and explain.
(f) One of the steps uses the fact that differentiable functions are continuous. Identify it and explain.