I came across this simple expression while doing some maths.
If
Then
Is this correct? How do we prove it?
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I came across this simple expression while doing some maths.
If
Then
Is this correct? How do we prove it?
Seems to be correct.
by definition.
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We start with
The chain rule says that for any two functions and ...
Applying the chain rule:Quote:
Originally Posted by Chain Rule
By our definition, this is equivalent to .
So . Done.
great work. thanks.
Wow, I realized that I had indeed overcomplicated it. Going back to edit again...
p.s. All in all, just know that the same basic rules of differentiation apply to unspecified functions. You'll just be left with more unspecified functions in your answer.
Since we're dealing with functions of , you'll be using the chain rule a lot.
etc.
Hey Dr., just curious...
So the basic rules of differentiation (product rule, constant multiple, chain, etc.) are general formulae derived via the difference quotient. Doesn't that mean that any function that can be differentiated with them can also be done directly by the difference quotient? (though certainly not as easily in general)
All of differential calculus can be derived from the definition of the derivative. Throw in the definition of the integral and you can develop the whole enchilada.
In fact all of mathematics is derivable from the Zermelo Fraenkel axioms, plus the Axiom of Choice and the various definitions. That is what mathematics is all about.
Of course, thanks.
I was asking if a function that can be differentiated by basic plug-and-chug rules (product, constant multiple, chain, etc.) can also be derived directly from a difference quotient, since the difference quotient is what gives rise to those rules. However it was a rather odd question, since the particular difference quotient makes for the definition of the derivative. So it's obviously true, as you pointed out.
Another question to continue... Can we go even further in saying that one can pretty much derive all elementary calculus simply from the limit? It seems one need only to apply the concept of the limit in a geometric sense to define the derivative of a curve, and likewise for the definite integral via Riemann sums.
A limit is nothing but a definition. That definition is used in the definition of both derivative and the integral.
There are very few actual assumptions in mathematics. In a formal sense those assumptions are the Zermelo-Fraenkel axioms of set theory plus the Axiom of Choice.
On a slightly less formal basis one can use the Peano axioms in place of the Zermelo-Fraenkel axioms. Both are in essence nothing more than the assumption of the natural numbers (0,1,2,3,4,...) . From that small set of assumptions, with some definitions one can construct the integers, the rational numbers, the real numbers and the complex numbers. From there you can get a great deal of mathematics. Add in the axiom of choice (given a collection of non-empty sets there is a function from that collection to their union that assigns to each set a member of the set) and all of the rest of mathematics (with the exception of elements of logic that study this process in an abstract fashion) follows.
If you want to see the various number systems developed from the Peano axioms, Landau's little book Foundations of Analysis is the place to look (there are other sources as well but this one is very direct and a very thin book). But bewarned, it is telegraphic and dry, something to be read once. You can read it over a short weekend. Also note that there are other ways to do the same thing that Landau does in his book -- he shows the development of the real number systems using Dedekind cuts, but it can also be done using equivalence classes of Cauchy sequences, the point being that in mathematics it is common for there to be different ways to approach something that result in the same end product even though the paths taken to reach that end can appear to be quite different.
You do not really derive mathematics, but rather construct it. The difference is that in constructing a mathematical theory one uses insight and imagination to make definitions that turn out to be useful. From those definitions, the basic axioms, and the consequences of those axioms and definitions made previously, one then applies logic to find the consequences.
Bottom line: You can do almost all of modern mathematics in many different ways, but ultimately it all boils down to a very few rather simple assumptions and the application of logic to some useful definitions.
You have seen only the tip of the iceberg. Elementary calculus is only a beginning and really it serves to give you an intuitive feel for the branch of mathematics called "analysis". More advanced analysis uses an extension of the Riemann integral, called the Legesgue-Steltjes integral that is able to handle more fuctions and to handle limiting operations that the Riemann iintegral cannot. Even the notion of the derivative can be generalized, to functions of several variables by using linear algebra and to functions that are not even continuous using the theory of Laurent distributions. You are several years of hard study away from these subjects.