I don't get how the derivative of the exponential function is derived.
This is the point where I start to not get.
(why does = )
The value of is the derivative of at
Assuming that is the value of at which the slope of the tangent line at is 1. (Why assume e?)
I get the rest.
I got a question for the differentation method of as well.
Guys, there is a reason I typed the "old fashioned" way of differentiating . I'm not looking for proofs of the formula, or other ways to solve it. I'm just looking to have my questions in brackets beside the steps answered (and the limit question on the bottom).
Thus, we end up with .
The real problem at that point is to know (or show if need be) that
If that must be shown it is difficult. From what you have written it looks like you are trying to follow the proof in James Stewart's textbook. That is one of the best proof's of this.
Notice that he defines as the number such that .
Does that help at all?
Any web link available for me to see the discussion?
EDIT: nevermind, I found it on Google Books. It is the same thing as Kumon, except explained in more depth, with graphs (Kumon "magically" expects you to understand this, or they expect you to spend time in Kumon with the professor so they can shrink the number of worksheets)
The good thing is the university level topics are not explored in ridiculous depth. I guess those are not for catching up school, but more for curious minds (like me) to get an idea of the material taught in university, and ease your way. For example, Kumon doesn't teach multi variable calculus, or inverse trigonometry. Linear algebra section only teaches matrix, mapping, and transformations.
And for those reasons, It is quite rare to find a student that finishes Kumon. Most people quit after a while because they get too frustrated either doing the overly challenging work, or frustrated from not understanding.