Definition of derivative

In summary, the concept of "differentiable" for a function from Rn to Rm can be extended to any place where there is a notion of linear map or a map such as |?| to the reals, or some other ordered space. This can be seen by defining "derivative" as a linear function from Rn to Rm with a remainder term that approaches 0 as the input approaches 0. In the case of a function from R1 to R1, the derivative is a number, but for other cases, it is a linear function or vector. This definition also allows for the calculation of partial derivatives and the use of the Hessian matrix to determine maxima, minima, and saddle points.
  • #1
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Saw this in the thread about "Explaining topology..."

matt grime said:
in R^n f is differentaible at x if there is a linaer map Df(x) satisfying

f(x+h)=f(x)+Df(x)h + ho(|h|)

this can be extended to any place where there is a notion of linear map or a map such as |?| to the reals, or some other ordered space.

Is there an easy way to extend this particular definition to when f : R^n -> R^m?
 
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  • #2
A function whose range is R^m is simply m functions whose range is R.
 
  • #3
Yes, that makes sense. What I'm getting at is the term proportional to h at the end. If h is a vector in R^n and f(h) is a vector in R^m, how do you make that particular definition work?
 
  • #4
Bleh, I wasn't paying any attention at all. I'm embarassed to have written my previous post!

That definition already works for f : R^n → R^m.


Though, I guess it's much less complicated to write R(h) instead of h o(|h|), and require that |R(h)| / |h| → 0. (R for remainder)
 
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  • #5
I could live with R(h) as the remainder if it is a vector in R^m. But it seems that h o(|h|) is a vector in R^n, and you can't add that to two vectors in R^m, can you?
 
  • #6
You're assuming the thing in the o(|h|) is a number: it could be, for example, an operator R^n → R^m whose operator norm is asymptotically less than |h|.
 
  • #7
You're right, I was assuming that. I know it doesn't really matter what the order is, but it might be more suggestive to write it o(|h|)h or something like that, to make it more clear that o(|h|) is a linear map. Thanks for the help :smile:
 
  • #8
It can't be a linear map. (Unless it's the zero operator)
 
  • #9
Ah, okay. Can you explain?

edit: I guess it's because, if it were linear, you could always add more vectors to the sum so that

o(|h|)(Sum of vectors) > |h| as h->0?
 
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  • #10
Well, if o(|h|) was hiding a linear map, then the remainder would look like:

R(h) = A h

for some matrix A...
 
  • #11
Right. I guess what I had in mind was a linear map parametrized by |h|, if that makes any sense at all. But that wouldn't be a linear map since it operates differently on different choices of h. Thanks.
 
  • #12
Of course, before you can define "derivative" of a function from Rn to Rm, you have to define "differentiable" (that's different from calculus I where a function is "differentiable" as long as the derivative exists!).

If f(x) is a function from Rn to Rm, the f is differentiable at x= a if and only if there exist a linear function, L, from Rn to Rm, and a function ε(x), from Rn to Rm, such that

f(x)= f(a)+ L(x-a)+ ε(x) and [tex]lim_{|x-a|->0}\frac{\epsilon}{|x-a|}= 0[/tex].

If that is true, then it is easy to show that the linear function, L, is unique (ε is not). We define the "derivative of f at a" to be that linear function, L.

Notice that, by this definition, in the case f:R1->R1, the derivative of f at a is a linear function from R['sup]1[/sup]->R1, not a number! However, any such linear function must be of the form L(x)= ax- multiplication by a number. That number is, of course, the "Calculus I" derivative of f.

Similarly, the derivative of a "vector valued function of a real variable", R1->Rm, is a linear function from R1 to Rm. Any such function can be written L(x)= x<a1, ...,am>, or x times a vector. That vector is the vector of derivatives in the usual "calculus III" sense.

The derivative of a "real valued function of several real variables", Rn->R1, is a linear function from Rn to R1. Such a function can be written as a dot product: <a1,...,an> dot product the x-vector. That vector is precisely the "gradient vector" of f. (And recall that, in Calculus III, a function may have partial derivatives at a point but not be "differentiable" there.)

This is, by the way, where the "second derivative test" for max or min (or saddle point) of a function of two variables comes from: You look at [itex]\frac{\partial^2F}{\partial x^2}\frac{\partial^2F}{\partial y^2}- \(\frac{\partial^2F}{\partial x \partial y}\)^2[/itex]. If that is negative at a (where the partial derivatives are 0), then there is a saddle point at a. If that is positive, then you have either a max or min depending on the sign of the second partials (which must be the same).

The point is that, if F:R2->R, then its derivative, at each point, can be represented as a 2-vector (the gradient vector). That means that the derivative function, that to each point assigns to that point the derivative vector, is a function from R2 to R2- and its derivative is a linear transformation from R2 to R2- which can be represented by a 2 by 2 matrix at each point (the "Hessian" matrix). The calculation [itex]\frac{\partial^2F}{\partial x^2}\frac{\partial^2F}{\partial y^2}- \(\frac{\partial^2F}{\partial x\partial y}\)^2[/itex] is simply the determinant of that matrix. Since the mixed second derivatives are equal, that matrix is symmetric and can, by a coordinate change, be written as a diagonal matrix having the eigenvalues on the diagonal. In that coordinate system, the equation for F is just ax2+ b2= C (no xy term) so if a and b are both positive we have a minimum, if both positive a maximum, if one positive, the other negative, a saddle point. Of course, the determinant (which does not change with a change of coordinate system) is just the product ab.
 
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  • #13
to define a derivative you first define what it means for a map to have derivative equal to zero. a map o(h) has derivative equal to zero at h =0 if and only if |o(h)|/|h| approaches zero as h does.

this makes sense for vector valued maps if |h| is a norm.

then the original definition makes sense (corrected) if we say that f is differentiable at x provided there exists a linear map L(h) such that the difference

f(x+h) - L(h) - f(x) has derivative equal to zero at h = 0.

then Df(x) = L.

in the original definition there is possibly an error, since the term h.o(h) should have been merely o(h) in this sense.
 
  • #14
So what precisely does little-oh notation mean when applied to a vector-valued function? I've only ever really used it in the context of algorithmic analysis, and time complexity isn't vector valued. :biggrin:
 
  • #15
exactly what i said: |o(h)|/(h| approaches zero ( a number), as |h| does, or equivalently as h does (a vector).
 
  • #16
Well, |o(h)| / |h| can't approach a number because, technically, o(h) is a set. :tongue2:

I guess this is what I assumed, so it's good to hear confirmation:

[tex]
f \in o(g) \Leftrightarrow \lim \frac{|f|}{|g|} = 0
[/tex]
 
  • #17
?

o(h) is a vector valued function of h, which is also a vector.

oh i see, my legg is being pulled..
 
  • #18
?

o(h) is a vector valued function of h, which is also a vector.

oh i see, my legg is being pulled.. this notation was used this way by hjardy and also by loomis, among others.
 
  • #19
!

In computer science, the notation o(f) is the class of all functions that are asymptotically less than f. Specifically:

[tex]
o(f) := \left\{ g \, | \,
\lim_{x \rightarrow \infty} \frac{g(x)}{f(x)} = 0
\right\}
[/tex]

(P.S. how do I get a nice, tall [itex]|[/itex]?)

I understand that here we are looking at x &rarr; 0 instead, but still...


If I were to say something like:

f(x) = x ln x + o(x)

what I really mean is that there exists some function g in o(x) such that

f(x) = x ln x + g(x)
 
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  • #20
Hurkyl said:
(P.S. how do I get a nice, tall |?)
Use: \left| and \right| ..... [tex] \left| \frac{\mbox{Big}}{\mbox{Expression}} \right| [/tex]
 
  • #21
It won't complain if it's already in the middle of a left/right pair? Or that it doesn't have a matched right to go with it?
 
  • #22
Nope. o:)

[tex] \left\{ \mbox{another} \left| \frac{\mbox{big}}{\mbox{expression}} \right\} [/tex]
 
  • #23
Thanks for the help, everyone. I think the definition that was given by Halls and mathwonk was probably what matt meant to write. Thanks for clearing that up. :smile:
 
  • #24
either that, or else he meant something different by the notation o(h). I.e. it would also suffice if he meant that notation to be a function that approaches zero as h does.

such functions were called infinitesimals in the older literature.
 

What is the definition of derivative?

The derivative of a function is the rate of change of that function at a specific point. It represents the slope of the function at that point and can also be interpreted as the instantaneous rate of change.

What is the difference between a derivative and a derivative function?

A derivative is a single value representing the rate of change at a specific point, while a derivative function is a function that gives the derivative value at every point along the original function. In other words, a derivative function is the collection of all the derivatives of a function.

How is the derivative of a function calculated?

The derivative of a function is calculated using the limit definition, which involves taking the limit of the slope of a secant line as the two points get closer and closer together. Alternatively, it can be calculated using differentiation rules, such as the power rule or chain rule.

What is the purpose of finding the derivative of a function?

The derivative of a function is useful in many applications, including physics, engineering, and economics. It can be used to find maximum and minimum values, determine rates of change, and analyze the behavior of a function.

What are the different notations used to represent derivatives?

The three most commonly used notations for derivatives are prime notation (f'(x)), Leibniz notation (dy/dx), and Lagrange's notation (Df(x)). Each notation has its own advantages and is used in different contexts.

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