How to differentiate 1/(x^3) using First Principle

lim h-->0 (1/ (x+h)^3 - 1/x^3) / h

Find common denominator: Note im not gonna keep writing limit...

(x^3 - (x+h)^3) / (h)(x+h)^3(x^3)

Foil out:
[x^3 - (x^3 + 3x^2h + 3xh^2 + h^3)] / [(h)(x+h)^3(x^3)]

Collect like terms:
[-3x^2h - 3xh^2 - h^3] / [(h)(x+h)^3(x^3)]

Cancel an h:
lim h-->0 [-3x^2 - 3xh - h^2] / [(x+h)^3(x^3)]

Take limit as h-->0
[-3x^2] / (x^3)(x^3)

[-3x^2] / (x^6)

Cancel an x^2:
[-3] / (x^4)

-3/x^4 <-- answer

Check by direct differentiation.
d/dx x^-3 = -3x^-4 = -3/x^4 :)