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.
Since
(how?)
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).
In your OP the limit is incorrect.
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?
It went on in Kumon as well (only for another page), but I didn't type it because I get it.
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)
A math (and reading, but known for math) tutoring program that's known to teach little kids basic elementary math but also has material all the way to calculus, linear algebra, and statistics (but almost nobody makes it there. One word sums up Kumon's curriculum below calculus, brutal)
Unforunately it is not. The content taught below calculus is way too difficult. The way they taught it is kind of "minimalistic" (thus requiring a really bright mind to understand if no one helped you, even though it is a "self taught" program). It makes perfect sense when you get it and look back, but not when you first learn it. It also goes into ridiculous depth (way more than school) that you won't ever need (factorization and optimization being the worst offenders).
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.