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As the title implies, It is seems that $e^x$ is the only function whoes derivative is the same as itself.

thanks.

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No, the derivative of $f(x) = 0$ is also itself :p (actually both are members of $ae^x$.) –  KennyTM Aug 22 '10 at 10:58
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In terms of what? In my view this is the defining property of $e$. –  anon Aug 22 '10 at 10:58
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Like muad, this is for me the definition of e^x. So I'm not sure where this intuition is supposed to be coming from. –  Qiaochu Yuan Aug 22 '10 at 11:39
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@Qiaochu: even if this is the definition of e^x for you, is it not then interesting that this e^x function can be characterized as the same as raising a particular constant between 2 and 3 to the power of x? Such a coincidence could warrant an intuitive explanation. –  Niel de Beaudrap Aug 22 '10 at 13:15
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Now that i think about how it was taught back then, it was quite a while in between deducing the properties of $\ln$ and $\exp$ ab initio, and then revealing that they are in fact the logarithm and exponential that was taught in previous courses! –  J. M. Aug 22 '10 at 13:23
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20 Answers

up vote 27 down vote accepted

Well, think of exponential growth (like e.g. bacteria grow):

We know, the more bacteria exist in a colony, the faster the colony will grow. More precisely: The growth speed of the colony $B$ is proportional to it's size ... Double size, double speed.

$$\frac{dB}{dt} \sim B$$

Furthermore we know the growth is exponential, since bacteria clone themselves in fixed amounts of time, i.e.

$$B \sim 2^{k\cdot t}$$

Putting it together, we can deduce that:

$$\frac{d}{dt}2^{kt} = c \cdot 2^{kt}$$

or with $a = 2^k$

$$\frac{d}{dt}a^t = c \cdot a^t$$


Now, how do we get the $e$? We just ask: What base $a$ do we have to take such that $c = 1$, i.e. $\dot{B} = B$?

We simply call that base $e$. Having such an $e$ is quite useful. We could use its special derivation traits we found above to define all exponential functions to the base $e$.

$$a^x = e^{x \cdot \ln a} $$

This shows that the factor $c$ we encountered in the above equations equals $\ln a = \log_e a$ and therefore, we can easily derive all kinds of exponential terms.

After all, $e$ turns out to be $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x \approx 2.718\ldots$

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An easily understandable explanation to laypersons is 'the more you have the faster it grows', like debt or savings (they're both compound interest but 'normal' people don't like fancy math terms ;) ). –  Marcin Jun 7 '11 at 12:07
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"The more bacteria exist in a colony, the faster the colony will grow" is false. Competition for resources adds a negating term which depends on the concentration of bacteria: $dB/dt = B - f(B)$. This makes all of the difference. –  user02138 Aug 14 '13 at 16:06
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(Looks like Niel and "J" beat me to the punch while I was composing this, but I'll post my answer anyway since I like my diagram. :) )

You can't get too much more intuitive than the Law of Exponents ($a^x a^c = a^{x+c}$) right?

To reduce symbolic (and mental) clutter for a while, let's write $k$ for $a^c$, to get

$$k \; a^x = a^{x+c}$$

The point is that

Multiplying the value of $a^x$ by something yields the same result as adding some other thing to the exponent.

Consider what this bit of algebra tells us about the geometry of the graph of the function $y=a^x$ (shown in red in the figure below). Recall some fundamental notions:

  • $y=k a^x$ is the result of vertically stretching (scaling) the original graph by a factor of $k$.

  • $y = a^{x+c}$ is the result of horizontally sliding (translating) the original graph by a (signed) distance of $-c$.

The Law of Exponents tells us that (provided $k$ and $c$ are related appropriately), these transformed graphs are identical! In the figure, the blue graph represents both results.[*] Importantly, what points move where aren't the same under both actions; scaling moves the red point $P$ vertically onto the blue point $Q$; translating moves the red point $R$ horizontally onto $Q$.

Diagram of Exponential Functions

Now, as suggested by the diagram, imagine a tangent vector poking out of each point of the original graph, and consider what the transformations to do those vectors:

  • Vertically stretching scales (only) the "rise" of the vector by factor $k$; that is, a vector with slope $m$ is pulled to achieve a slope of $m k$.
  • Horizontally translating has no effect on the vector's slope.

What can we conclude here? Why, something pretty remarkable:

For any point $P$ on the original graph, the point $R$ --located $c$ units to the right[**] of $P$-- is such that the slope of the tangent vector at $R$ is $k$-times the slope of the tangent vector at $P$ (where $k$ and $c$ are related appropriately).

Let me take this opportunity to retire $k$, since it is beginning to become clutter; I'll just make the appropriate relationship explicit. Also, I'm going to retire the point name $R$, opting to describe the point instead. I'll summarize things this way:

On the graph $y=a^x$, if the slope of tangent vector at any $P$ is $m$, then the slope of the tangent vector at the point $c$ units to the right of $P$ is $m\;a^c$.

Notice that there's nothing special about the players in this game. $P$ is any point on the graph, $c$ is any (horizontal) distance you care to choose; heck, even the exact value of $a$ is up for grabs. Let's use this to our advantage.

Suppose we take $P$ to be the point where the (original, red) graph crosses the $y$-axis; that is, we take $P$ to have $x$-coordinate $0$ (and $y$-coordinate $1$, but this doesn't really matter). Then, "the point $c$ units to the right of $P$" can be described more simply as "the point with $x$-coordinate $c$", and we have

On the graph $y = a^x$, if the slope of the tangent vector at the point with $x$-coordinate $0$ is $m$, then the slope of the tangent vector at the point with $x$-coordinate $c$ is $m\;a^c$.

In Calculusian prose:

If $f(x) = a^x$, then $f^{\prime}(c) = a^c f^{\prime}(0)$.

Observe that we don't even need "$c$" in the above formula, since it's just taking the place of some $x$-coordinate. We can simply write:

If $f(x) = a^x$, then $f^{\prime}(x) = a^x f^{\prime}(0)$.

From here, it's pretty much a matter of definition to get to the final answer to your specific question. After all, the above holds for any (non-negative) value of $a$. Clearly, some values of $a$ correspond to graphs that cross the $y$-axis very steeply; some values correspond to graphs that cross the $y$-axis very shallowly; it's not un-reasonable to believe that there's a convenient value would cause the graph to cross the $y$-axis juuuuuuuuust right ... with a slope of $1$. Of course (as I mix my folklore), to name a demon is to control him, so we'll simply assign a symbol to this "just right" value of $a$.

Let "e" be the number such that the tangent vector to $y=e^x$ at $x=0$ has slope $1$.

Assuming that such a number really does exist, you don't even have to know its exact value to conclude

If $f(x) = e^x$, then $f^{\prime}(x) = e^x$.

You can then turn your attention to figuring out why $e$ happens to have the value $2.718...$ . Other answers in this thread provide insights on how those arguments proceed.

Also left as an exercise is to determine why, in the formula for the derivative of $a^x$, the "$f^{\prime}(0)$" factor is in fact "$\log a$" (the natural logarithm of $a$).

[*] As Niel mentions, this illustrates that the graph of an exponential function is "self-similar". I think it's important to add "uni-directionally" to the description, in order to distinguish it from (conventional) similarity transformations where scaling occurs "omni-directionally".

[**] "to the left" works, too, with appropriate changes to the discussion.

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At least yours has color, Don. :) –  J. M. Aug 22 '10 at 22:05
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Doesn't quite explain $e^x$, but looking at the series of $2^n$, i.e. $\{1,2,4,8,16,32,...,2^k,2^{k+1},...\}$ might help.

You see there if you take the first order difference you get back the same series, as $2^{k+1}-2^k=2^k$.

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Therefore, the function $2^x$ is such that the secant slope over a "unit run" is $2^x$, which invites the question: What about over smaller runs? Let $a$ be such that the secant slope of $a^x$ over a run of $h$ is $a^x$; that is, suppose $(a^{x+h}-a^x)/h=a^x$. Dividing through by $a^x$, we solve to get $a=(1+h)^{1/h}$. Taking $h$ smaller and smaller pushes us toward a value of $a$ such that the secant slope over an "infinitesimal run" --that is, the tangent slope-- is $a^x$. Of course, this is the same limit we see over and over again, but it arises from a fairly intuitive motivation. –  Blue Aug 23 '10 at 11:26
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As J.M. says,

$$ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!} + \cdots $$

If you accept (it's true, but it could take a while to explain why) that this series, like polynomials, can be derived term by term

$$ p(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \ \Longrightarrow \ p'(x) = a_1 + 2a_2x + \cdots + na_n x^{n-1} \ , $$

then

$$ \frac{d}{dx} e^x = \frac{1}{1!} + \frac{2x}{2!} + \cdots + \frac{nx^{n-1}}{n!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x \ . $$

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The approach I would have taken though, was to express $\exp(x)$ first as 1+(other terms), take the derivative so that you now have the general term $\frac{n x^{n-1}}{n!}=\frac{x^{n-1}}{(n-1)!}$, and shift indices accordingly. –  J. M. Aug 22 '10 at 11:49
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Mmh, isn't the infinite sum obtained by using a tailor series that follows the observation that $f^{(N)}(0) = f(0) = 1$? Kinda circular explanation ... –  Dario Aug 22 '10 at 11:56
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Well, this is the formulation assuming that the power series is the definition of $\exp(x)$. One now shows that this power series satisfies the differential equation in the title. –  J. M. Aug 22 '10 at 12:00
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Could this power series be the definition of $\exp(x)$, since itself is based on the fact that the derivative of $\exp(x)$ is $\exp(x)$? –  Jichao Aug 22 '10 at 16:23
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@Jichao. For a lot of books, this series is the definition of $e^x$. –  a.r. Aug 22 '10 at 18:06
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One conceptual way to understand the standard definition $e^x = \lim_{n\to\infty}(1+x/n)^n$ is to view it as arising from applying Euler's approximation method to $\,y' = y.\,$ The same method also works for arbitrary higher-order constant coefficiant ODEs by converting them to linear system form and employing matrix exponentials. This is described quite nicely in Arnold's beautiful textbook Ordinary differential equations. Here is an excerpt:

alt text alt text

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Thanks for showing the picture of the Euler polygon! In the Hairer-Norsett-Wanner book, the picture of the Euler polygons slowly converging to the true solution is referred to as "Lady Windermere's fan", after the character by Oscar Wilde. –  J. M. Aug 22 '10 at 15:10
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What does E mean here? –  Casebash Aug 22 '10 at 21:21
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Casebash: Reckoning from the formula for Euler's method, it's apparently the identity matrix. –  J. M. Aug 23 '10 at 4:44
    
Euler method can be used efficiently with an adaptative step. –  Felix Marin Oct 1 '13 at 4:13
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I'll act on Casebash's proposal later, but for now... care for a movie?

tangent line

(granted, I cheated and spaced the two plots of $\exp(x)$ to clearly show the moving tangent line... but you guys should be able to understand this).

(thanks to Stan Wagon!)

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Suppose $\frac{d}{dx}f(x)=f(x)$.

Then for small $h$, $f(x+h)=f(x)+hf(x)=f(x)(1+h)$. If we do this for a lot of small intervals of length $h$, we see $f(x+a)=(1+h)^{a/h}f(x)$. (Does this ring a bell already?)

Setting $x=0$ in the above, and fixing $f(0)=1$, we then have $f(1)=(1+h)^{1/h}$, which in limit as $h\rightarrow 0$ goes to $e$. And continuing $f(x)=(1+h)^{x/h}$, which goes to $e^x$.

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Fair warning: The rigorous formulation can be found in any textbook. What I am giving here is strictly informal intuition behind the result. If you write this in your exam you will get a zero :) However, this can be made filled out and rigorous with a lot of arguments –  KalEl Aug 22 '10 at 11:45
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Well, this is not really an intuition but an heuristic for the proof of the statement. –  Mariano Suárez-Alvarez Aug 22 '10 at 13:25
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A particle on a trigonometric hyperbola has acceleration described by the same hyperbola.

That is, differentiation is idempotent on the hyperbolic cosine. Since $e^{x} = (\cosh x)^{\prime} + \cosh x$, the claim follows immediately.

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I thought this was a nice argument. No love? –  user02138 Nov 23 '10 at 23:28
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This is pretty cool! I think this answer is just lost in the sea of (good) answers, and relatively few users venture down this far. –  Jesse Madnick Aug 14 '13 at 20:10
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One interesting way of looking at this is by turning the differential equation into an integral equation.

The problem $ y'=y $ is the same as $ \int y \ dx = y $. For the problem to be well defined I have to define an initial condition, we use the condition $y(0)=1$.

Now I am going to guess an answer for y. The simplest function $y(x)$ that I know which goes through $(0,1)$ is $y_0(x)=1$. The subscript here has no special mathematical meaning, it just signifies that this is my first guess.

If we put our guess into the integral equation we get, $$ \int y_0(x) \ dx = \int 1 \ dx = x + C . $$

The integral added a constant that wasn't there before, our guess must have not been very good so we will take this new result as our second guess hoping that it knows more about the problem. So our new guess is,

$$y_1(x) = x + C = x + 1 $$

Notice that we must set C=1 so that $y_1(0)=1$. We will substitute this into the integral equation again which will give us a new guess $y_2(x)$.

$$ y_2(x) = \int y_1(x) \ dx = \int x + 1 \ dx = \frac{x^2}{2} + x + C = \frac{x^2}{2} + x + 1 $$

If we do this a few more times we will get the following,

$$ y_3(x) = \frac{x^3}{6} + \frac{x^2}{2} + x + 1 $$

$$ y_4(x) = \frac{x^4}{24} + \frac{x^3}{6} + \frac{x^2}{2} + x + 1 $$

If you look closely at the coefficients you will see that we are generating the Maclaurin series for $e^x$.

This is happening because the only function which is it's own anti-derivative is the exponential function. In the terminology of dynamics we would say that the infinite series representation of $e^x$ is a fixed point of the integral operator. It is also an attractor which means that repeated integration of $some$ unrelated functions will produce a sequence of functions that approach $e^x$.

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Let us define a function $\exp$ as the solution to $D \exp(x) = \exp(x)$ with initial value $\exp(0) = 1$

Clearly $D \exp \circ f = D f \cdot (\exp \circ f)$ but also notice that $$ \begin{align} D (\exp(a(x)) \exp(b(x))) &= D a(x) \exp(a(x)) \exp(b(x)) + D b(x) \exp(a(x)) \exp(b(x)) \\ &= (D (a + b) (x)) \cdot \exp(a(x)) \exp(b(x)) \end{align} $$ so we see that $\exp(a)\exp(b)=\exp(a+b)$ which lets us write $\exp(x) = e^x$ for some as yet undetermined constant $e$.

Geometrically we can see that the gradient of the tangent line of the curve $y = e^x$ must be between the finite differences, i.e. $\frac{e^x - e^{x-h}}{h} < e^x < \frac{e^{x+h} - e^x}{h}$. With some numerical exploration we see that $e$ is between $2$ and $3$. Specifically, the finite differences for $e=2$ and $h=0.9$ are too small and the finite differences for $e=3$ and $h=0.1$ are too big. To narrow it down more let $e=2.7$ and $h = 0.01$ to see that $2.7 < e < 3$.

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$d/dx (e^{-x} \cdot y)=-e^{-x} \cdot y+e^{-x} \cdot dy/dx=e^{-x}(dy/dx-y)$

So if $dy/dx=y$, then $d/dx (e^{-x} \cdot y)=0$, ie $e^{-x} \cdot y=c$ or $y=c \cdot e^x$

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Can someone remind me of the name of this method? –  Casebash Aug 22 '10 at 11:12
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Doesn't this assume $d(e^{-x})/dx = -e^{-x}$ in the first place? –  KennyTM Aug 22 '10 at 11:13
    
@Kenny: Yes, but given that it is a solution, it shows why it is the only solution. Maybe it isn't as intuitive as Jichao would like, but I think its a good start –  Casebash Aug 22 '10 at 11:15
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Suggestion: \cdot rather than * for multiplication. –  anon Aug 22 '10 at 11:22
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There are two components to the original question:

  • Why do exponential functions generally (the class $x \mapsto ab^x$ for $b>0$) have the property of being eigenfunctions of the derivative operator? That is: why do these functions f have the property $\frac{\mathrm d}{\mathrm dx} f(x) = \lambda f(x)$?

  • Why are the functions $x \mapsto a \mathrm e^x$ in particular +1 eigenfunctions --- i.e. why is the scalar $\lambda$ equal to 1 for these functions in particular? (To put it another way: why is this the case when the base is equal to the particular constant e ≈ 2.71828?)

I can give an intuitive answer to the first of these questions, by combining two observations.

  1. Consider the slope at a paticular point, e.g. at x = 0. Multiplying the function f(x) by some scalar now only amplifies the function by that factor, but also amplifies the slope by that same factor. (This is just a restatement of the product rule, for constant scalar factors.)

  2. Exponential functions f(x) are self-similar: multiplying them by a positive scalar is equivalent to translating them to the left, or to the right, depending on the scalar by which you multiply: more precisely, if you multiply $\mathrm e^x$ by c, why you get is $\mathrm e^{[x + \ln(c)]}$, or the same function translated ln(c) to the left. More generally, for $f(x) = ab^x$, we have cf(x) = f(x + logb(c)); that is, multiplying f(x) by c is equivalent to a shift to the left by logb(c).

Now, combine these two facts. The slope of f(x) at x = logb(c) is c times the slope of f(x) at x = 0, because of the self-similarity of the exponential function, and because the slope of cf(x) at x = 0 is c times the slope of f(x) at x = 0. Thus the slope of f(x) is a function which is proportional to f(x), QED.

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This is the rigorous way of stating what the animation in my answer was showing. –  J. M. Aug 22 '10 at 13:26
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$cf(x)=cf(x+logb(c))$ should be $cf(x)=f(x+logb(c))$. –  Jichao Aug 22 '10 at 16:41
    
@Jichao: indeed, thanks for spotting the typo. Fixed now. –  Niel de Beaudrap Aug 22 '10 at 21:12
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[This is taken from "Mathematics of the Discrete Fourier Transform", by Julius Smith, and anon's answer, possibly others, are also similar. I like the intuitiveness of Smith's presentation though].

Derivatives of $f(x) = a^x$.

Apply the definition of differentiation:

$$f'(x_0) = \lim_{\delta \to 0}{\frac{f(x_0 + \delta) - f(x_0)}{\delta}}$$ $$= \lim_{\delta \to 0}{\frac{a^{x_0 + \delta}-a^{x_0}}{\delta}}$$ $$= \lim_{\delta \to 0}{a^{x_0}}\frac{a^\delta - 1}{\delta}$$ $$= a^{x_0} \lim_{\delta \to 0}\frac{a^\delta - 1}{\delta}.$$

Since the limit of $(a^\delta -1) / \delta$ as $\delta \to 0$ is less than 1 for $a = 2$ and greater than 1 for $a = 3$ (by direct calculations), and since $(a^\delta -1)/\delta$ is a continuous function of a for $\delta > 0$, it follows that there exists a positive real number we'll call $e$ such that for $a=e$ we get $$\lim_{\delta \to 0}\frac{e^\delta - 1}{\delta} = 1.$$

For $a=e$, we thus have $$(a^x)' = (e^x)' = e^x.$$

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In 'simple terms', we can define the ${\large{\rm e}}$ number as the one which makes 'simple' the logarithm derivative:

$$ {{\rm d}\log_{a}\left(x\right) \over {\rm d}x} = \lim_{h \to 0}{\log_{a}\left(x + h\right) - \log_{a}\left(x\right) \over h} = \lim_{h \to 0}{1 \over h}\,\log_{a}\left(1 + {x \over h}\right) = {1 \over x}\,\lim_{h \to 0}\log_{a}\left(1 + {x \over h}\right)^{x/h} $$

'Simple' means that we take, as a definition, $$ a = \lim_{h \to 0}\left(1 + {x \over h}\right)^{x/h} \quad\mbox{which we call}\quad {\large e}\quad\mbox{and}\quad \ln \equiv \log_{\rm e} $$

It establishes the 'simple' relation $$ {{\rm d}\ln\left(x\right) \over {\rm d}x}= {1 \over x} $$

Then $$ \ln\left({\rm e}^{x + y}\right) = x + y = \ln\left({\rm e}^{x}\right) + \ln\left({\rm e}^{y}\right) = \ln\left({\rm e}^{x}\,\,{\rm e}^{y}\right) \quad\Longrightarrow\quad {\rm e}^{x + y} = {\rm e}^{x}\,\,{\rm e}^{y} $$

Intuitively, it's hard. Somebody told me that Euler was tired of the accumulation of logarithm powers when he took successive derivatives of $a^{x}$.

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When we take $x^n$, and derive it $n$ times, we get $n!$ . So, by that logic, if we'd ever want to construct a function (other than the obvious $f(x) = 0)$ which is immune to derivation (and, by extension, to integration as well, since the two are opposite operations), then that function would have to be $\sum_0^\infty\ x^n/n!$ . Which “just-so-happens” to be the same as $e^x$ , since the famous number $e$ has been expressly created, designed, and defined to equal exactly $\sum_0^\infty\ 1/n!$ , precisely for this very reason.

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start from $f_0(x)=1$ but then $f'_{0}(x)=0$ we need something more, let's try $f_1(x)=1+x$ but then $f_1'(x)=1$ we need something more, let's try $f_2(x)=1+x+\dfrac{x^2}{2!}$ then $f'_2(x)=1+x$, continuing this way we see that $f_n(x)-f_n'(0)=\dfrac{x^n}{n!}$

But for large enough $n$, $\dfrac{x^n}{n!}\approx0$, and that is good enough.

Form the finitist point of view there is no function $f_n(x)$ s.t. $f_n(x)-f_n'(0)=0$ So $e^x=\dfrac{\mathrm de^x}{\mathrm dx}$ is really abuse of notation.

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If $y'(x)=y(x)$, then $y''(x)=y'(x)=y(x)$, $y'''(x)=y''(x)=y'(x)=y(x)$, and so on: all derivatives are equal.

Assuming that $y(x)$ admits a Taylor expansion, it will be:

$$y(x)=y(0)+y'(0).x+y''(0).x^2/2+y'''(0).x^3/3!+...)$$ $$=y(0)+y(0).x+y(0).x^2/2+y(0).x^3/3!...$$ $$=y_0(1+x+x^2/2+x^3/3!+...).$$ You recognize the Talyor expansion of $y_0.e^x.$

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Assume $y(0)=y_0$, and estimate $y(x)$ by stepping in small increments $h=\frac{x}{n}$.

By Taylor, we develop the first order approximation $y(h)\approx y(0)+y'(0).h=y(0)+y(0).h=y_0(1+h)$.

Then $y(2h)\approx y(h)+y'(h).h=y_0(1+h)^2$.

Then $y(3h)\approx y(2h)+y'(2h).h=y_0(1+h)^3$.

And more generally, $y(nh)\approx y_0(1+h)^n$.

This shows that the function grows exponentially: every time you make a step, you multiply by the same factor $(1+h)$.

We just established that $y(x)\approx y_0(1+\frac{x}{n})^n$, taking $n$ very large.

In the limit, you recognize the exponential function.

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Consider the closely related finite difference version: $\Delta y_n=y_n$.

We have $\Delta y_n=y_{n+1}-y_n=y_n$, so that $y_{n+1}=2.y_n$, an exponential behavior.

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Here's my 2 cents, just look at the def of derivative

df/dx = lim f(x+d)-f(x)/d

now the defining property of ex is that for very small x, ex ~ 1 + x

and since ex + d = exed ~ ex(1+d), it's clear that by plugging everything into the definition of derivative you will get the required result.

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I think we can make these ideas rigorous with nonstandard analysis. (That is, interpret the ~ to be standard part equality and 'very small' to mean infinitesimal). –  anon Aug 22 '10 at 21:22
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I would not say that $e^x \sim 1+x$ is a defining property of $e^x$... –  Bruno Joyal Dec 14 '11 at 8:51
    
@Bruno The answer was not elaborated enough, but I disagree: For x small, $e^x \sim 1 + x$ is a defining property of $e$, and a pretty good one. I've tried to give a more elaborated version of why in my answer. –  Greg Hill Jun 19 '13 at 16:38
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