What is the derivitive of 1?

The derivative of a constant is always 0. To show this, let's apply the definition of derivative.

Recall that the definition of derivative is:

f'(x) = lim h→0 (f(x + h) - f(x))/h

Let f(x) = 1. Then:

f'(x) = lim h→0 (1 - 1)/h

= lim h→0 0/h

= lim h→0 0

= 0!