Differentiation from first principles

Differentiaiton from principles requires the use of the following formula which is provided in the formula booklet:

f'(x) = lim_h->0 ((f(x+h) - f(x))/(h))

Consider a function:

f(x) = 6x²

Clearly we know that the function differentiates to:

f'(x) = 12x 

by using the process of multiplying the coefficient by the power and then reducing the power by 1.

Using first principles however we must consider the formula mentioned previously.

f'(x) = lim_h->0 ((f(x+h) - f(x))/(h))

By computing the function for x+h and x we get:

f'(x) = lim_h->0 (6(x+h)² - 6x²)/(h))

f'(x) = lim_h->0 (6(x²+2xh+h²) - 6x²)/(h))

f'(x) = lim_h->0 (6x²+12xh+6h²) - 6x²)/(h))

f'(x) = lim_h->0 (12xh+6h²)/(h))

We now cancel the h from above and below to get:

f'(x) = lim_h->0 12x+6h

Now consider the limit as h-> 0, clearly 12x remains unaffected but 6h will become 0 and is hence removed. Hence we are left with:

f'(x) = 12x

Which we know to be true from the trivial methods of differentiation considered earlier.