Existance of Slope

The slope of f at a does not always exist. Here is a list of all the possibilities.

(1)  The slope of f at a exists if the ratio

is finite and has the same standard part for all infinitesimal Δx ≠ 0. It has the value

(2)    The slope of f at a can fail to exist in any of four ways:

(a)    f(a) is undefined.

(b)    f(a + Δx) is undefined for some infinitesimal Δx ≠ 0.

(c)    The term is infinite for some infinitesimal Δx ≠ 0.

(d)    The term has different standard parts for different infinitesimals Δx ≠ 0.

We can consider the slope of f at any point x, which gives us a new function of x.

The derivative f'(x) is undefined if the slope of f does not exist at x.

For a given point a, the slope of f at a and the derivative of f at a are the same thing. We usually use the word "slope" to emphasize the geometric picture and "derivative" to emphasize the fact that f is a function.

The process of finding the derivative of f is called differentiation. We say that f is differentiable at a if f'(a) is defined; i.e., the slope of f at a exists.