Derivative: $e^x$. How do you differentiate $e^x$?  I looked on many sites, including similar questions here but most answers seemed circular.  The only known definition of $e$ to be used in this proof is  $$ e=\lim_{n \to\infty} \left(1+\frac{1}n \right)^n $$  What I did is:  $$ \begin{align*} (e^x)' &amp;=\lim_{h\to0}\frac{e^{x+h}-e^x}h \\ &amp;= e^x\lim_{h\to0}\frac{e^{h}-1}h \end{align*} $$  But I don't know how to go on, I know  $\lim_{h\to0}\frac{e^{h}-1}h=1$ but I don't know how to prove it, I can't use the $e^x$ taylor expansion as that would imply diferentiating $e^x$.  Edit: I also can't use the derivative of $\ln(x)$.
Using the Limit definition to find the derivative of $e^x$ I was wondering how we could use the limit definition   $$ \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$  to find the derivative of $e^x$, I get to a point where I do not know how to simplify the indeterminate $\frac{0}{0}$. Below is what I have already done  $$\begin{align} &amp;\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\ &amp;\lim_{h \rightarrow 0} \frac{e^{x+h}-e^x}{h} \\ &amp;\lim_{h \rightarrow 0} \frac{e^x (e^h-1)}{h} \\ &amp;e^x \cdot \lim_{h \rightarrow 0} \frac{e^h-1}{h} \end{align}$$  Where can I go from here? Because, the $\lim$ portion reduces to indeterminate when $0$ is subbed into $h$.
I know that if we solve the Maxwell equation, we will end up with the phase velocity of light being related to the permeability and the permittivity of the material. But this is not what I'm interested in - I want to go deeper than that. We know that the real speed of light is actually not changing, the decrease in speed is just apparent. Material is mostly empty, the light will still travel with $c$ in the spacing. The rare atoms will disturb the light in some way. So I am interested in how the atoms affect the light. Photon absorption-emission theory Some textbooks that I read explain it in a way kind of like this: In a material the photons are absorbed by atom and then re-emitted a short time later, then they travel a short distance to the next atom and get absorbed&amp;emitted again and so on. How quickly the atoms in a material can absorb and re-emit the photon and how dense the atoms decides the apparent speed of light in that material. So the light appears slower because it has a smaller “drift speed”. Interference theory But recently I realize an alternative explanation: Atoms respond to the light by radiating electromagnetic wave. This “new light” interferes with the “old light” in some way that results in delayed light (advanced in phase), this can easily be shown by using simple phasor diagram. Consequently effectively the light covers a smaller phase each second, which gives the impression of a lower phase velocity. However the group velocity is changing in a complicated way. I think that the first explanation does not explain the change in phase velocity of light. if we consider light travelling into a slab of negative refractive index non-dispersive material, let’s say the light is directed perpendicular to the slab. The phase velocity’s direction will be flipped, but group velocity’s direction in the material will not change. Only the second explanation can explain the flipped phase velocity direction. I guess that the velocity that we get in the first explanation is actually belongs to the group velocity. It makes sense to me that the front most of the photon stream determines the first information that the light delivers. So the question is What really cause the phase velocity of light to be decreased?  &quot;drift velocity&quot; of photons (they aren't the same photons, they are re-emitted all the time) phase difference between absorbed and emitted light something else  And also, I still don't really understand detailed explanation of the absorption-emission process for small light's wavelength (for large lambda compare to the atoms spacing, the photons will be absorbed by the phonons). The dispersion relation that we know is continuous and also some material is non-dispersive, therefore the absorption process must occur in all frequency for a certain range. So definitely it doesn't involve the atomic transition, otherwise it will be quantized. My guess is that the relevant absorption process gets smooth out by the dipole moment. What makes the spectrum continuous? EDIT: link for dispersion relation: Unable to mount NTFS external hard drive How to Make This Block Diagram with LaTeX I understand that there are some questions here regarding the block diagrams. However, I am strongly interested in one particular style shown as below:    This is a picture from a paper. As can be seen, the text can be selected there. So it is not an attached picture. I suppose it is done by LaTeX.  Could anyone point me to the right direction to achieve this?  (I specially LOVE the colored "thin grid"! It is much more beautiful than the fully filled color!) How to extract and join files xxx.zip, xxx.z01 and xxx.z02 I have three files 1.zip, 1.z01 1.z02. I would like to extract and join them into a file, by right clicking 1.zip and selecting "Extract Here". But then there comes an error:     7-Zip 9.04 beta  Copyright (c)   1999-2009 Igor Pavlov  2009-05-30   p7zip Version 9.04   (locale=en_US.utf8,Utf16=on,HugeFiles=on,2   CPUs)      Error: /home/tim/Desktop/1.zip:   Can not open file as archive      Errors: 1   I was wondering what the problem it is and how I can solve it?  Thanks and regards!    ADDED: Is my usage of "unzip -FF" wrong? It seems not work:  $ unzip -FF 1.zip  Archive:  1.zip warning [1.zip]:  zipfile claims to be last disk of a multi-part archive;   attempting to process anyway, assuming all parts have been concatenated   together in order.  Expect "errors" and warnings...true multi-part support   doesn't exist yet (coming soon). file #1:  bad zipfile offset (local header sig):  0   Thanks again!  2nd ADDED: Still ain't right.  $ cat 1.z01 1.z02 1.zip &gt; combined.zip &amp;&amp; unzip -FF combined.zip Archive:  combined.zip warning [combined.zip]:  zipfile claims to be last disk of a multi-part archive;   attempting to process anyway, assuming all parts have been concatenated   together in order.  Expect "errors" and warnings...true multi-part support   doesn't exist yet (coming soon). warning [combined.zip]:  209829313 extra bytes at beginning or within zipfile   (attempting to process anyway) file #1:  bad zipfile offset (local header sig):  209829313   (attempting to re-compensate)  extracting: 1.wmv   bad CRC ee181eef  (should be f3c61875)   It generated 1.wmv, which can play well only until  midway in Gnome Mplayer.  3rd ADDED: Am I using 7z wrongly:  $ 7z e 1.z01  7-Zip 9.04 beta  Copyright (c) 1999-2009 Igor Pavlov  2009-05-30 p7zip Version 9.04 (locale=en_US.utf8,Utf16=on,HugeFiles=on,2 CPUs)  Processing archive: 1.z01  Error: Can not open file as archive  $ 7z e 1.zip  7-Zip 9.04 beta  Copyright (c) 1999-2009 Igor Pavlov  2009-05-30 p7zip Version 9.04 (locale=en_US.utf8,Utf16=on,HugeFiles=on,2 CPUs)  Processing archive: 1.zip  Error: Can not open file as archive I want to get random decimal digits out of a 6-faced die.  Let's say the die is fair, with an equal probability for each outcome. (But feel free to share any consequences of that not being true, if you feel like it.)  I will represent the outcome of a die roll as a base-6 number:   In base-6 there are only 6 possible digits: $\{0,1,2,3,4,5\}$. The die has faces numbered 1 to 6. Face $f$ corresponds to digit $f-1$.   Thus, I can represent as many die rolls as I like with a base-6 number:   If the die is rolled once:   Let's say the die stopped on face $5$. The final base-6 number will be $4$.  If the die is rolled twice:   Let's say the die stopped on face $5$. And then on face $3$. The final base-6 number will be $42$.  If I roll the dice 10 times:   Let's say the die stopped sequentially on faces $5$, $3$, $4$, $6$, $2$, $3$, $3$, $3$, $2$, and $4$. The final base-6 number will be $4235122213$.    Since I want decimal digits, I need to convert to base-10.   Base-6: $4235122213$ Base-10: $44753985$   For any given number of die rolls, how many different combinations exist? If there are $r$ die rolls, the total of possible combinations is $6^r$.   $r$ rolls — $6^r$ different combinations. $0$ rolls — only $1$ combination. $1$ roll — $6$ different combinations. $2$ rolls — $36$ different combinations. $3$ rolls — $216$ different combinations. $4$ rolls — $1296$ different combinations. $5$ rolls — $7776$ different combinations. $6$ rolls — $46656$ different combinations. $7$ rolls — $279936$ different combinations. $8$ rolls — $1679616$ different combinations. $9$ rolls — $10077696$ different combinations.   $n_r$ ($\in \mathbb{N_0}$) is the number obtained after $r$ die rolls through the process described previously: "If the die is rolled once, twice, ..."  It is impossible for $n_r$ to reach the number of possible different combinations ($6^r$). It will be always short by least $1$:  $$ 0 \leqslant n_r &lt; 6^r $$   $r$ rolls — $n_r$ never surpasses $6^r-1$. $0$ rolls — $n_r$ never surpasses $0$. $1$ roll — $n_r$ never surpasses $5$. $2$ rolls — $n_r$ never surpasses $35$. $3$ rolls — $n_r$ never surpasses $215$. $4$ rolls — $n_r$ never surpasses $1295$. $5$ rolls — $n_r$ never surpasses $7775$. $6$ rolls — $n_r$ never surpasses $46655$. $7$ rolls — $n_r$ never surpasses $279935$. $8$ rolls — $n_r$ never surpasses $1679615$. $9$ rolls — $n_r$ never surpasses $10077695$.   I can never trust the leftmost nor the rightmost digit (fringe digits), since the possible values are potentially restricted.   For $0$ rolls, the outcome is always $0$. For $1$ roll, the fringe digit never surpasses $5$. For $2$ rolls, the leftmost digit never surpasses $3$, and the rightmost, $5$. For $3$ rolls, the leftmost digit never surpasses $2$, and the rightmost, $5$. For $4$ rolls, the leftmost digit never surpasses $1$, and the rightmost, $5$. For $5$ rolls, the leftmost digit never surpasses $7$, and the rightmost, $5$. For $6$ rolls, the leftmost digit never surpasses $4$, and the rightmost, $5$. For $7$ rolls, the leftmost digit never surpasses $2$, and the rightmost, $5$. For $8$ rolls, the leftmost digit never surpasses $1$, and the rightmost, $5$. For $9$ rolls, the leftmost digit never surpasses $1$, and the rightmost, $5$.   The other digits seem perfectly random to me though. Am I wrong? I've been living in Ireland for almost a year now and I start noticing they use the word "mens" a lot. I can see it used in:   Shops, to denote the area where you can find men's clothes In sport, when they talk about "mens team".    My guess is just that they are lazy about the use of quotes, so that mens should actually be men's. However, there may be some rule I'm not aware of. So, is "mens" only limited to Irish English? When I'm allowed to use it? Seeking English term for relationship between two members of extended family? Here at the Genealogy and Family History Stack Exchange we see many questions about relationship mapping which is also known as:   relationship calculation kinship terminology etc   If you are just trying to understand how to calculate "cousins and removes" then  provides an excellent guide.  However, if you are seeking an English term for a more complicated relationship between two members of an extended family, rather than asking a question like:          site users are encouraged to first review the answers below.    Note: This Q&amp;A arises from  and is set to  to help make it canonical. What's the best way to protect your house from Endermen stealing your walls? Suppose that there is a trigonometric equation of the form $a\sin x + b\cos x = c$, where $a,b,c$ are real and $0 &lt; x &lt; 2\pi$. An example equation would go the following: $\sqrt{3}\sin x + \cos x = 2$ where $0&lt;x&lt;2\pi$.  How do you solve this equation without using the method that moves $b\cos x$ to the right side and squaring left and right sides of the equation?  And how does solving $\sqrt{3}\sin x + \cos x = 2$ equal to solving $\sin (x+ \frac{\pi}{6}) = 1$ Default Class Accessibility in C# by default is a class:   private ? internal ?  sealed ? Continuous footnote numbering I would like footnote numbering to continue over pages and chapters. In my document it's reset every chapter. I'm using the report documentclass. Seeking Mobile Data collection software recommendations? When users create a node, they should be redirected to a payment page; if payment is successful, the user can see the node.  Is there any way to achieve this? Is there a route that would allow travel to the US from the UK or Schengen area, given the US travel ban? I'm a European citizen living in the US on a work visa. Due to some unfortunate circumstances, I need to urgently visit my home country. However I would like to get back to the US after my visit is complete. Flying from the US to Europe is not a problem as there are still daily flights between some city pairs. However going back could be tricky, as  anyone who has been in the Schengen area in the past 14 days from re-entering the country.   Is there a flight route that would let me from Europe to some country, stay there for 14 days and then fly to the US? Canada is out, as they've banned international visitors. Mexico seems to keep their border open, but I  any direct European flights from their airports. Almost every other country either suspended flights or doesn't allow foreign nationals to enter. How can I test the quality of a RNG? On electronics.stackexchange we had  about constructing a True Random Number Generator. Since the method relies on noise, which isn't deterministic, the only way to test the quality of the RNG seems to be empirical. Not being a statistician I suggested to test a long bit sequence for normality, but I have no idea what outcome is acceptable and what isn't. For instance counting single bits I guess we have a thumbs-up for a 499500/500500 distribution, but when is the ratio too much skewed to be acceptable? Same for 2-bit and longer sequences. Of course, if testing for normality is a Bad Idea™, I'd like to hear it, including better alternatives.    edit Diehard was mentioned a few times, but I'm not sure this answers my question. The normality test should give a normal distribution, with better bell curve approximations for longer sequences. Diehard seems also to have tests which either should result in normal or exponential distributions. But my question remains: how do I judge the results? Just by looking at the curve and discern a bell curve in it? To get back at my first example, a 499500/500500 distribution is definitely OK, and a 950000/50000 distribution definitely is a no-no, so where does the switch-over happen? overset text over equal sign in in an Align environment Look at the following example  \begin{align}      &amp; x \\      \overset{\text{some long explanation}}= &amp; foo \\      &amp; = bar \end{align}   what is the best way to get the equal signs aligned? Now I have seen different solution such as adding \overset{\hphantom{\ţext{some long explanation}}}= to the beginning of every line, but I don't really think that is the best solution. Any better ideas?    UPDATE  I guess what i am actually after is   aligning the first column of the align environment right aligning the equal sign under the overset text right extracting energy from cosmological expansion This question is a more concrete reincarnation of an .  Are there mechanisms to extract energy from the cosmic rate of expansion? putting some extremely long ropes across the universe and extract work from their increase of separation?  assuming this is possible ( i don't see why it would not ), what form would take the back-reaction force? in all physics, every time you extract work from some system by some force, this system will exert a force back. Can we slow down cosmic expansion by variations of these mechanisms? Unzip File: Permission Denied I am trying to unzip a file (Facebook download-your-info zip file), but here's what I got:   Is there a way I can force-unzip it?  Update: I tried sudo unzip file.zip -d destination_folder in terminal, and it unzipped the file, but what came out, all the files have size 0 bytes:   Update 2: I do have permission to the directory and the file.    (I'm really sorry if my question is asked badly. Let me know down in the comments :) ) Why are my sauces splitting? Recently all my attempts at sauces are splitting and I'm not sure what's going wrong. Tonight I tried a balsamic vinegar reduction / chicken stock / pan juices recipe to go with my lamb chops, but once again the dreaded split occurred. My wife is patient and forgiving, but I feel like I must be missing something fundamental to get this wrong so often.  Two questions.   What could I be doing wrong to cause these sauces to split? What can I do to recover when it happens again?   UPDATE: The Pan juices contained a tbsp or two of olive oil, and some of the fresh herb rub I'd coated the lamb chops in (thyme, rosemary, and mint). I did not drain the pan before starting the sauce. It has crossed my mind that perhaps there was too much oil / fats in the pan at that point?  For completeness, I should point out that I also sauteed some chopped shallots in the pan juices immediately before adding the vinegar. MacPorts fails to find 'make' I've installed Xcode and MacPorts on a brand new mid-2013 MacBook Air, but MacPorts doesn't work, saying that it cannot find anything.  mba:~ cnst$ sudo port install wget Warning: The Command Line Tools for Xcode don't appear to be installed; most ports will likely fail to build. Warning: See http://guide.macports.org/chunked/installing.xcode.html for more information. ---&gt;  Computing dependencies for wgetError: Unable to execute port: can't read "build.cmd": Failed to locate 'make' in path: '/opt/local/bin:/opt/local/sbin:/bin:/sbin:/usr/bin:/usr/sbin' or at its MacPorts configuration time location, did you move it? mba:~ cnst$   And, indeed, there is no make or clang in the path, but it is definitely installed, though:  mba:~ cnst$ find /Applications -name "clang" -or -name "make" /Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/clang /Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/lib/clang /Applications/Xcode.app/Contents/Developer/usr/bin/make mba:~ cnst$   Why is there no make or clang in the path?  Why MacPorts doesn't work?  I notice that  mentions that you have to install something from ADC, Apple Command Line Developer Tools, but ADC doesn't work since last Thursday or Friday due to their security breach, so, the developer tools are nowhere to be found.  What's the proper solution here?