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In Thurston's philosophical paper, "On Proof and Progress in Mathematics", Thurston points out that mathematicians often think of a single piece of mathematics in many different ways. As an example, he singles out the concept of the derivative of a function, and gives seven different elementary ways of thinking about it:

(1) Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function.
(2) Symbolic: the derivative of $x^n$ is $nx^{n-1}$, the derivative of $\sin(x)$ is $\cos(x)$, the derivative of $f \circ g$ is $f' \circ g * g'$, etc.
(3) Logical: $f'(x) = d$ if and only if for every $\epsilon$ there is a $\delta$ such that when $0 \lt | \Delta x | \lt \delta, |\frac{f(x + \Delta x) - f(x)}{\Delta x} - d| \lt \delta$
(4) Geometric: the derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent.
(5) Rate: the instantaneous speed of $f(t)$, when $t$ is time.
(6) Approximation: The derivative of a function is the best linear approximation to the function near a point.
(7) Microscopic: The derivative of a function is the limit of what you get by looking at it under a microscope of higher and higher power.

Thurston goes on to state that "The list continues; there is no reason for it ever to stop," so let's keep it rolling! Can you come up with other ways of thinking about the derivative? I should remark that this is a list of different ways of thinking about the derivative, which isn't the same thing as a list of different formal definitions of the derivative. Remember to limit yourself to one answer per post.

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There's some here: golem.ph.utexas.edu/category/2009/08/… –  Matt May 17 '10 at 21:27
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Well, one way could be as here: abstrusegoose.com/26 –  Regenbogen May 17 '10 at 22:09
    
I saw the title and came in hoping to say "hey, you should read Thurston's paper!" but I see you already did. –  Michael Lugo May 17 '10 at 22:56
    
I feel like Thurston should have replaced "the limit of what you get by looking at it" with "the slope of the line you get in the limit by looking at the graph" in #7. –  S. Carnahan May 17 '10 at 23:24
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Trivial point: In (3), one of the deltas should be an epsilon. –  Gregory Putzel Aug 2 '12 at 2:49
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11 Answers

The intuitive idea is that the derivative is zero if there is a local minimum or maximum.

The derivative is then the (tangent of) the angle by which we have to rotate the graph around that point in order to get a local minimum or maximum.

You could probably cook up some sort of way to think about what happens when the function vanishes to odd order (first of all, no matter how you rotate it, there will not be a local minimum or maximum).

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An algebraic definition which works for at least polynomials: Consider the ring $\mathbb{R}[dx]/(dx^2)$ the derivative of a function $f(x)\in \mathbb{R}[x]$ could be defined implicitly by $$f'(x)dx +(dx^2) = f(x+dx)-f(x)+(dx^2).$$

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There are two closely related interpretations of the derivative of a generating function in combinatorics.

  • If $A(x) = \sum a_n x^n$ counts the number of $A$-structures on an $n$-element ordered set, then $A'(x) = \sum na_n x^{n-1}$ counts the number of ways to add a distinguished element to an $n-1$-element ordered set and to choose an $A$-structure on the result.

  • If $A(x) = \sum \frac{a_n}{n!} x^n$ counts the number of $A$-structures on an $n$-element set, then $A'(x) = \sum \frac{a_n}{(n-1)!} x^{n-1}$ counts the number of ways to add an element to an $n-1$-element set and to choose an $A$-structure on the result.

Actually in combinatorics it is more natural (in both setups) to consider $x \frac{d}{dx}$, which is referred to as "pointing." This opens up a lot of interesting ideas; for example, one can interpret certain differential equations combinatorially and try to find their solutions symbolically.

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I like a different geometric interpretation from the graphical one. If you think about a function f:R-->R as a transformation on the real line R, then the interpretation of f'(x) is that it's the scaling factor of this transformation near the point x.

This interpretation is good for a geometric understanding of the change-of-variables formula for integrals. It also makes the chain rule pretty plain.

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I actually see this as similar to (6). –  David Corwin Dec 28 '12 at 5:14
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For me the derivative is a measure of a function's sensitivity to a small change in its input. So it tells you, for any given input, what the ratio of the change in output over the a small change in input (which caused the change in output).

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In smooth infinitesimal analysis, one can prove that a curve can't be a set of points but can instead be built out of line segments whose lengths are nilsquare infinitesimals (different from the infinitesimals of NSA, which obey the transfer principle). The derivative is then simply the slope of one such segment.

A good intro that doesn't require any knowledge of category theory is Bell, A primer of infinitesimal analysis.

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Marsden & Weinstein use the "method of exhaustion" to define the derivative without limits.

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Nice . –  Martin Brandenburg May 18 '10 at 7:39
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The derivative of a Regular Type is its Type of One-Hole Context

This is a surprising computational application related to Qiaochu's example, but different enough to warrant some explanation. Apologies for some of my hand-waveyness.

In a suitably pure programming language (like Haskell) we can think of data types as being objects of a category and functions (suitably qualified) as arrows. We often want to build one type from another. For example, given a type $X$ we can form the type $X\times X$, the type of pairs of $X$'s. Similarly we can form coproducts which correspond to objects that can be of one type or another. Eg. if $Y=X+X^2$ then $Y$ is a type which contains either an $X$ or a pair of $X$'s. The functions that construct one type from another, call them type constructors, can naturally be thought of as functors. So $P$ defined by $P(X)=X^2$ is a functor. If $(x,y)$ is in $X^2$ then $Pf(x,y)=(f(x),f(y))$. Types form a semiring.

Sometimes we want to make a 'hole' in a type constructors. If $F$ is a type constructor, then $F(A)$ can be thought of as a container of elements of type $A$. An "$F$ with a hole in it" is one of these containers but with one of its elements removed, but still keeping information about where the element was removed from. For example consider $F(X)=X^3$. $F$ makes triples of $X$'s. A triple with a hole in it consists of just two $X$'s as well as enough information to tell where the third element was removed from. There are just three places it could have come from, so we can describe the removal site using a type with just three elements. Call it $3$. So a triple with a hole in it is a pair consisting of an element of type $3$ and an element of type $X^2$. Ie. $3X^2$. This is $F'(X)$.

This works more generally. We make holes in type constructors by differentiating them.

It even works with recursive types. For example, a list of $X$'s is, by definition, either the empty list, or a pair consisting of an $X$ and a list. We have an equation

$L(X)=1+XL(X)$.

We can differentiate this to get

$L'(X)=L(X)+XL'(X)$

This gives a recursive equation for a type of a list with a hole in it. This is in fact an example of a type called a zipper. Used in many places such as this application. In the particular case of lists $L'(X)$ defines a pair of lists of $X$'s.

(Much of this applies in the category $Set$ but the recursive equations can introduce some issues of well-foundedness if taken too literally.)

Many of the usual properties of derivatives acquire straightforward computational interpretations: linearity, the product rule, the chain rule, even the Faà di Bruno formula.

Anyway, check out the papers which have none of the errors I've probably introduced. There's also a close relationship with combinatorial species.

(Curiously, you can also make sense of finite differences of types even though we only have a semiring and don't have subtraction of types.)

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I think the answers so far do not really take account of Kevin's remark:

I should remark that this is a list of different ways of thinking about the derivative, which isn't the same thing as a list of different formal definitions of the derivative.

We could just copy from http://en.wikipedia.org/wiki/Derivative_%28generalizations%29 or cite Fermat categories .... Perhaps this is an alternative way:

Smoothing: Continuity in $p$ asserts roughly that you can draw the graph in one line. Differentiabilty is the next step towards smoothness: The line has no zig-zags / edges. The derivative is then the direction of the line.

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My answer was a way of thinking about the derivative: anything which is additive with a product rule can have all the intuitions and expectations of the derivative applied to it (maybe with some surprises in characteristic p). –  KConrad May 18 '10 at 1:13
    
Different definitions lend themselves to different ways of thinking, so this isn't so bad, I think. –  Qiaochu Yuan May 18 '10 at 1:19
    
Yes, but then you should formulate this different way of thinking. In many cases, it happens to be one already mentioned by Thurston. –  Martin Brandenburg May 18 '10 at 12:25
    
But the graph can have an infinite sequence of zig-zags/edges approaching $p$ and still be differentiable at $p$... –  Noam D. Elkies Aug 2 '12 at 8:08
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If $R$ is a ring, then a derivation $\partial : R \to R$ is a vector field over the scheme $\mathrm{Spec} R$.

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Can someone explain what's wrong with this answer? The words all sound vaguely reasonable to me, and I don't know enough algebraic geometry to understand why the sentence doesn't work. –  Vectornaut May 18 '10 at 19:45
    
This is true !! More precisely, if X=Spec A and if T is the tangent sheaf of X (over the base Z), as defined in EGA 4, then a global section of T is the same as a derivation of A. –  user2330 May 19 '10 at 9:06
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Algebraic: a derivation on a ring $R$ is an additive map $R \rightarrow R$ that satisfies the product rule (with suitable generalizations allowing modules, etc.)

This is related to the Symbolic way of thinking on Thurston's list, but is not identical to that (e.g., it leads to important characterizations of separability for field extensions).

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It makes sense to talk of derivations on semirings. (In fact, my answer is about an example.) Is there much of a developed theory of calculus on semirings. In particular, mutually recursively defined types look remarkably like varieties over semirings meaning we can mimic the construction of tangent bundles over semirings. Is there any developed theory for this? Maybe this should be a proper question. –  Dan Piponi May 18 '10 at 20:34
    
I don't know about that. –  KConrad May 18 '10 at 20:48
    
To me, this way of thinking seems more geometric than symbolic, because if R is the ring of smooth functions on a manifold, the product rule guarantees that derivations are local: if two functions agree on an open neighborhood of x, their derivations agree at x. Ever since I found that out, I've been wondering whether you can derive the product rule from linearity, locality, and some other natural condition (I don't think the first two are enough). Thoughts? –  Vectornaut May 20 '10 at 18:14
    
The space of derivations at a point can be identified with the dual space of M/M^2 (M being the maximal ideal of functions vanishing at the point), and from the special features of the manifold setting the mere linearity can indeed be pulled back to recover a derivation. Here's the setup. Let A be a local ring with maximal ideal M and set F = A/M. (Think A = local ring of smooth functions at point P on manifold, M = elements of A vanishing at P, F = real numbers). We want to show any F-linear map f : M/M^2 --> F can be pulled back naturally to a derivation A ---> F. More in next comment. –  KConrad May 20 '10 at 19:23
    
Let us make an assumption that is true in setting of manifolds but is not always true: the field F = A/M naturally lies inside the ring A. More precisely, suppose there is a homomorphism F ---> A such that the composite F --> A --> A/M is the identity. (For rings of functions at a point on a manifold this is very natural since the real numbers sit inside A as constant functions naturally. But not every local ring naturally contains its residue field, such as a local ring of char. 0 whose residue field has characteristic p.) We get a natural direct sum decomposition A = F (+) M. More next... –  KConrad May 20 '10 at 19:28
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