The slope of a function y(x) is the slope of its TANGENT LINE
Close to x=a, the line with slope y ' (a) gives a "linear" approximation

y(x) is close to y(a) + (x - a) times y ' (a)

If you want to solve y(x) = 0, choose x so that y(a) + (x - a) y ' (a) = 0
This is a really fast way to get close to the exact solution to y(x) = 0 :

"Newton's Method" x = a - y(a)/y '(a) SEE THE EXAMPLES