Question

how did newton find the slope of a tangent line to a function at some point

Answer 1

first principles

Answer 2

taking the limit of a function as h-->0 where h is the change in x (x2-x1)

Answer 3

He used his method of fluxions (his name for the derivative)!
Basically he calculated the limit:
\[ \lim_{h->0} \frac{f(x+h)-f(x)}{h} \]