If f(x) = x^3, then f(x + h) = (x + h)^3.

( f(x+h) – f(x) ) / h

= ( (x + h)^3 – x^3 ) / h …(1)

Expanding (x + h)^3 gives:

(x + h) * (x + h)^2

= (x + h) * (x^2 + hx + hx + h^2)

= (x + h)(x^2 + 2hx + h^2)

To do the next multiplication, you cannot use FOIL, as there are more than 2 terms in the right hand bracket.

The first bracket contains x and h.

So multiply the second bracket by x, then by h, and add the results:

x(x^2 + 2hx + h^2)

= x^3 + 2hx^2 + xh^2 …(2)

h(x^2 + 2hx + h^2)

= hx^2 + 2h^2x + h^3 …(3)

Now add (2) and (3):

(x + h)^3

= x^3 + 2hx^2 + xh^2 + hx^2 + 2h^2x + h^3

= x^3 + 3hx^2 + 3h^2x + h^3.

Substituting this in (1):

( f(x+h) – f(x) ) / h

= ( (x + h)^3 – x^3 ) / h

= (x^3 + 3hx^2 + 3h^2x + h^3 – x^3) / h

= (3hx^2 + 3h^2x + h^3) / h

= h(3x^2 + 3hx + h^3) / h

= 3x^2 + 3hx + h^3.