Would this closed form derivative rule for tetration be correct for $n$ as a nonnegative integer? If so, can it be more concise? For the recent week I've been diving myself into tetration out of pure curiosity, and after watching videos about how to find the derivative of $^2x$ and $^3x$, I decided that I wanted to look into what a derivative rule for it would be. When I searched online I couldn't find anything that was satisfying to me, so I put some work in and through a couple iterations of solutions, I think I have a closed form derivative rule I am personally satisfied with. What I found was: $$\frac{d}{dx}(^nx)=\frac{^nx}{x} \sum_{a=0}^{\lceil n \rceil -1} \ln(x)^a \prod_{b=0}^a \frac{\ln(^{n-b}x)}{\ln(x)}.$$ Firstly, I've been wanting to get on this website for a while, so I thought I would bring it here. Secondly, I wanted to know if this looks accurate to anyone here, as through my testing with it, it seems to be. And lastly, I'm just a Calc BC student, exploring into these further realms of mathematics myself because it interests me, so I don't know, given that this is accurate, if there is a more concise or elegant way to put this derivative rule together, so is there a way I could? A note about this is that I have it working for $n$ being any nonnegative real number, and for that to work, I am going by the recursive definition for tetration of: $$^nx:=\begin{cases} 1  &amp; \text{if } n = 0\cr x^{\big(^{(n-1)}x\big)} &amp; \text{if } n &gt; 0, \end{cases}$$ which extends it so that $n$ can be any nonnegative real number, such that $^{2.47}x$ would be considered $x^{x^{x^{0.47}}}$. Update: As I know that the recursive definition for tetration is not exactly the definition for tetration across the reals, and that that is still a researched topic, I wanted to share my derivative rule which I made before this one that works for $n$ as any nonnegative integer. Here is what I have on that: $$\frac{d}{dx}(^nx)=\frac{^nx}{x}\sum_{a=0}^{n-1}\ln(x)^a\prod_{b=0}^a{^{n-b-1}x}.$$ For this, I ask the same questions as I asked above. Thank you for your time reading this.
Finding the derivative of $x\uparrow\uparrow n$ I am trying to find a general derivative for the function: $f(x)=x^{x^{x^{...^{x}}}}$however to do that I must find $f^{\prime }$ and $f^{\prime \prime}$...etc. I am now trying to write down a general expression for $f^{\prime \prime}$ and I have stumble opon the series in the picture below, I wonder, do anyone have an idea of what sort of expantion it migth be ? I have tried breaking it down into three different sums, but I wasn't very successful!  EDIT  So since this post has gotten some attention and the title has been change I will tell you what I am on about, although I already have a post just for the question in the title from before. I am as I said earlier trying to find a general derivative for the function $f(x)=x^{x^{x^{...^{x}}}}$ where $n$ is an arbitrary number, natural number and greater or equal to 2. I have already made success when I tried to find the first derivative, and I have also found some interesting pattern emerge when the derivatives as well as n incresses. But my question here was if anyone knew what this expation might be or look similar to be, since I have yet to be lernt series etc. I did not know that the question needed such background since it's straight a forward question.
How to install the OS with a new Hard drive? Most Common Aspect Ratio? I'm building a web site that will allow users to upload photos. For design and layout purposes, I need to know the most common aspect ratios users will upload. What is the most common aspect ratio of photos (e.g from dSLR, smart phones, etc.)? How can I retrieve data within my widget I have a Widget which I am creating but my quesiton is how do I access my custom table which I created via my plugin. I am attempting to fill the dropdown from the table called tfp which contains the column liters which I wish to popluate the values into the drop down   class my_plugin extends WP_Widget {  // constructor function my_plugin() {         parent::WP_Widget(false, $name = __('Thomsons Fuels Widget', 'wp_widget_plugin') );  }  // widget form creation function form($instance) {  //Check values if( $instance) {  $title = esc_attr($instance['title']);  $dropdown= esc_attr($instance['dropdown']);  $textarea = $instance['textarea'];  } else {  $title = '';  $textarea = ''; } ?&gt;   &lt;p&gt; &lt;label for="&lt;?php echo $this-&gt;get_field_id('dropdown'); ?&gt;"&gt; &lt;?php _e('Liters',   'wp_widget_plugin'); ?&gt;&lt;/label&gt; &lt;select&gt; &lt;option value="volvo"&gt;Volvo&lt;/option&gt; &lt;/select&gt;  &lt;/p&gt;   How I created the table in the plugin   global $wpdb; global $custom_table_example_db_version;  $table_name = $wpdb-&gt;prefix . 'tfp'; // do not forget about tables prefix  // sql to create your table // NOTICE that: // 1. each field MUST be in separate line // 2. There must be two spaces between PRIMARY KEY and its name //    Like this: PRIMARY KEY[space][space](id) // otherwise dbDelta will not work $sql = "CREATE TABLE " . $table_name . " (   id int(11) NOT NULL AUTO_INCREMENT,   liters VARCHAR(100) NOT NULL,   price VARCHAR(100) NOT NULL,      postcodearea varchar(20) NULL,         dft varchar(20) NULL,   PRIMARY KEY  (id) );";  // we do not execute sql directly // we are calling dbDelta which cant migrate database require_once(ABSPATH . 'wp-admin/includes/upgrade.php'); dbDelta($sql);  // save current database version for later use (on upgrade) add_option('custom_table_example_db_version', $custom_table_example_db_version); I want a translucent PNG overlayed on top of my screen (above all windows). I'm using this to mark a visible region while streaming.   So far I've figured out that qiv -transparent filename.png will add a mask to the window, but it's a boolean mask. I don't get translucent fading with the alpha channel.  I know several effects have opacity, and I can sent window opacity, so I'm hoping there's some way to display an image that way.  What program, tool, or widget can I use to blend an image onto the top of the desktop? At least it's my understanding that AES isn't affected by known-plaintext. Is it immune to such an attack, or just resistant? Does this vary for chosen-plaintext? Force as gradient of scalar potential energy My text book reads      If a particle is acted upon by the forces which are conservative; that is, if the forces are derivable from a scalar potential energy function in manner $ F=-\nabla V $.   I was just wondering what may be the criteria for force to be expressed as negative gradient of scalar potential energy and HOW DO WE PROVE IT? Safety of installing Ubuntu alongside Windows Is it really safe to choose the "Install Ubuntu alongside Windows" option from the Ubuntu installation program?  I never used that option, instead I used other tools such as partition magic or windows 7's disk tool to resize my partitions.  Since I'm going to install Ubuntu across a lot windows (XP, Vista and 7) machines. It would not be nice to damage those Windows partitions.  What is your experience? Can I use this feature without concerns? Undercooked beans in chili? I made a bit of a mistake, being a lazy bachelor and inexperienced at cooking.  I was making chili and when it came time to add beans I discovered I didn't have any canned beans.  I did however have some dried beans.  I decided to save time and add these directly to the pot of chili and add some water.  However, after much simmering the beans maintain a certain unpalatable crispness.  I have also read that undercooked beans contain a chemical lectin which can cause stomach discomfort.  I have also read that beans should be soaked, the water discarded to remove the lectin, and then boiled to cook them.  Is there any way to salvage this?  Should I just continue cooking the chili for hours?  Will I have to throw out my potentially delicious chili? How do I study the "correlation" between a continuous variable and a categorical variable? What gear do I need for a newborn photography session? I am an amateur photographer and I have a newborn session coming up for a family friend.   My current gear is:   Canon T5i (crop sensor) Canon 18-55 IS STM Kit lens 85mm f/1.8 lens   I plan on buying a reflector for the shoot and will be using natural light as my light source.  So my question is: is this enough gear to get some nice photos? Should I get another prime lens besides the 85mm? I don't really use the kit lens.  I also have another 8 month old baby shoot coming up and an informal wedding shoot coming up so it would be nice to use the lenses in all situations. I don't need a telephoto at the moment and will probably not use one any time soon. Relationship between Binomial and Beta distributions I'm more of a programmer than a statistician, so I hope this question isn't too naive.  It happens in sampling program executions at random times. If I take N=10 random-time samples of the program's state, I could see function Foo being executed on, for example, I=3 of those samples. I'm interested in what that tells me about the actual fraction of time F that Foo is in execution.  I understand that I is binomially distributed with mean F*N. I also know that, given I and N, F follows a beta distribution. In fact I've verified by program the relationship between those two distributions, which is  cdfBeta(I, N-I+1, F) + cdfBinomial(N, F, I-1) = 1   The problem is I don't have an intuitive feel for the relationship. I can't "picture" why it works.  EDIT: All the answers were challenging, especially @whuber's, which I still need to grok, but bringing in order statistics was very helpful. Nevertheless I've realized I should have asked a more basic question: Given I and N, what is the distribution for F? Everyone has pointed out that it's Beta, which I knew. I finally figured out from Wikipedia () that it appears to be Beta(I+1, N-I+1). After exploring it with a program, it appears to be the right answer. So, I would like to know if I'm wrong. And, I'm still confused about the relationship between the two cdfs shown above, why they sum to 1, and if they even have anything to do with what I really wanted to know. Polynomial $P(a)=b,P(b)=c,P(c)=a$ Let $a,b,c$ be $3$ distinct integers, and let $P$ be a polynomial with integer coefficients.Show that in this case the conditions $$P(a)=b,P(b)=c,P(c)=a$$ cannot be satisfied simultaneously.  Any hint would be appreciated. How to add electrical fixtures to (custom) furniture? Having electrical outlets, USB charging ports, night lights, etc would be nice to have on some pieces of furniture, like side tables. Several receptacle manufacturers make combination fixtures with USB and power, for instance, or power plus a night light.   I've thought about nailing an electrical box to a (hidden) portion of the table and putting one of those receptacles in it, but what things do I need to consider doing this? Is it as simple as it sounds,  and then just fashioning some kind of plug out of 12/2 or 14/2 wire? Does the answer change if the box is concealed (though still accessible) vs it being completely visible? What are the meaning and possible origin of "word!" and "word up"? Several times, I have had conversations, all over instant messenger, finish with "word" or "Word up G".  As it ends a conversation, I am guessing it is like "goodbye".  My question is what is the meaning of "word" and "word up g"? Also, what is the origin?  I am more interested in the meaning as that will help with understanding its usage. Half doesn't or half don't? I'm having some issues SSH'ing to a server when logged into a particular VM.  From my local PC, I can log in to the server just fine using the command ssh -i .ssh/[my private key] username@domain.  If I then SSH into another unrelated server which also has the same private key and run the above command it will also log in just fine in to the server.  But, when I SSH into a particular OpenStack VM (again has the same private key) on the unrelated server and then SSH in to the server from there I start getting prompted for a passphrase. But a passphrase hasn't been set for the key so inevitably I get a "Permission denied (publickey)." error.  To double check, I removed the private key from the VM and scp'd the private key from my PC into it again and tried the same command I still get prompted for a passphrase.  In other words:   PC - SSH - Server1 = Works fine  PC - SSH - Server2 - SSH - Server1 =     Works fine  PC - SSH - Server2 - SSH - Virtual Machine - SSH - Server 1 =         Asks for a passphrase   What could be the reason behind this? It's definitely the right private key I'm using and it has the appropriate permissions.  Edit - Output of ssh -v -i .ssh/[my private key] username@domain as requested  OpenSSH_7.4p1, OpenSSL 1.0.2k-fips  26 Jan 2017 debug1: Reading configuration data /etc/ssh/ssh_config debug1: /etc/ssh/ssh_config line 58: Applying options for * debug1: Connecting to xxx.xxx.xxx.xxx [xxx.xxx.xxx.xxx] port 22. debug1: Connection established. debug1: key_load_public: No such file or directory debug1: identity file /.ssh/[my private key].ppk type -1 debug1: key_load_public: No such file or directory debug1: identity file /.ssh/[my private key].ppk-cert type -1 debug1: Enabling compatibility mode for protocol 2.0 debug1: Local version string SSH-2.0-OpenSSH_7.4 debug1: Remote protocol version 2.0, remote software version OpenSSH_7.2p2 Ubuntu-4ubuntu2.4 debug1: match: OpenSSH_7.2p2 Ubuntu-4ubuntu2.4 pat OpenSSH* compat 0x04000000 debug1: Authenticating to xxx.xxx.xxx.xxx:22 as '[my username]' debug1: SSH2_MSG_KEXINIT sent debug1: SSH2_MSG_KEXINIT received debug1: kex: algorithm: curve25519-sha256@libssh.org debug1: kex: host key algorithm: ecdsa-sha2-nistp256 debug1: kex: server-&gt;client cipher: chacha20-poly1305@openssh.com MAC: &lt;implicit&gt; compression: none debug1: kex: client-&gt;server cipher: chacha20-poly1305@openssh.com MAC: &lt;implicit&gt; compression: none debug1: kex: curve25519-sha256@libssh.org need=64 dh_need=64 debug1: kex: curve25519-sha256@libssh.org need=64 dh_need=64 debug1: expecting SSH2_MSG_KEX_ECDH_REPLY debug1: Server host key: ecdsa-sha2-nistp256 SHA256:ZiugTjR5fM0E3evOwHoePFKspDQChA0Ab6L0q88KP/g debug1: Host 'xxx.xxx.xxx.xxx' is known and matches the ECDSA host key. debug1: Found key in /home/centos/.ssh/known_hosts:2 debug1: rekey after 134217728 blocks debug1: SSH2_MSG_NEWKEYS sent debug1: expecting SSH2_MSG_NEWKEYS debug1: SSH2_MSG_NEWKEYS received debug1: rekey after 134217728 blocks debug1: SSH2_MSG_EXT_INFO received debug1: kex_input_ext_info: server-sig-algs=&lt;rsa-sha2-256,rsa-sha2-512&gt; debug1: SSH2_MSG_SERVICE_ACCEPT received debug1: Authentications that can continue: publickey debug1: Next authentication method: publickey debug1: Trying private key: /.ssh/[my private key].ppk Enter passphrase for key '/.ssh/[my private key].ppk': Enter passphrase for key '/.ssh/[my private key].ppk': Enter passphrase for key '/.ssh/[my private key].ppk': debug1: No more authentication methods to try. Permission denied (publickey). Why doesn't planet Earth expand if I accelerate upwards when standing on its surface? According to General Relativity I am being accelerated upwards by planet earth while writing this question. But a curious person on the the other side of the planet relative to me would have the same experience. That means we are accelerated in opposite directions, although earths diameter do not seem to increase. How can this be? How would I be able to get N results for several groups in an oracle query.  For example, given the following table:  |--------+------------+------------| | emp_id | name       | occupation | |--------+------------+------------| |      1 | John Smith | Accountant | |      2 | Jane Doe   | Engineer   | |      3 | Jack Black | Funnyman   | |--------+------------+------------|   There are many more rows with more occupations.  I would like to get three employees (lets say) from each occupation.  Is there a way to do this without using a subquery? Environment.TickCount vs DateTime.Now When should a verb be followed by a gerund instead of an infinitive?