Proof for first principle equation in derivatives?

Swetasuria

Please explain how this equation is derived.

f'(x)= lim [f(x+h)-f(x)]/h
h→0

Thanks.

micromass

It's a definition, it cannot be derived or proven.

Maybe your question should be, "why did who choose this particular definition", or "what is the intuition behind this definition". The answers to these questions are bound to be imprecise though. Is that what you want to ask?

Swetasuria

Yes. Also I'm not so clear by the idea of a derivative. Please help.

SteveL27

Have you seen the picture of the graph of a function, with the slope of the line between two points on the graph?

If one x-value is x and the other is x+h, then there are two points on the graph: (x, f(x)) and (x+h, f(x+h)).

Then the slope of the line passing through these two points is

(f(h+x) - f(x)) / h

It's just the old "difference in y over difference in x" method of computing the slope of a line, right?

Now if you hold x fixed and let h go to zero, you get the limit of the slope as the two points move closer together. That's the geometrical idea behind the definition of the derivative.

I'm thinking this picture must be in your text, since it's pretty standard.

Swetasuria

But what do you mean by the limit of the slope?

SteveL27

Imagine the two points on the graph of a function. You hold one point fixed; and you let the other point "roll" toward the first point.

For each position of the moving second point, you can calculate the slope of the line through the two points. If the moving slope gets arbitrarily close to some value, we call that value the limit. Didn't you study limits in class before getting to derivatives?

genericusrnme

In my highschool they taught derivatives before limits...
I'd be willing to bet that it's probably done more often than it should be.

Swetasuria

Yeah, we do but I never really got it.

So, does derivative of a function mean the limit of its slope?

SteveL27

Well, the "limit of the slope" idea is the geometric intuition for the definition of the derivative. It's a helpful picture to keep in mind as you work with derivatives.

In other words we define the derivative as the limit of the difference quotient; but we secretly think about the rolling secant line (the line between two points on the graph). That's the motivation behind the definition.

It's perfectly normal to have difficulties understanding limits. Mathematicians struggled for a couple of centuries to arrive at the modern notion of limit. If you include Archimedes, who certainly had the right intuition about infinite processes, you could say that mathematicians struggled for literally two thousand years to get to our modern ideas.

The essential idea of a limit is that some variable quantity gets arbitrarily close to some other quantity. That vague idea has to suffice until the student eventually takes a course in real analysis, where they make everything logically rigorous. But in beginning calculus, you have to take some of this on faith. It's troubling to ask students of mathematics to take things on faith; but that's the way calculus is taught.

If you post specific problems you're having trouble with, people can definitely help. And the more problems you do, the more the ideas sink in and become familiar.