I'm asked to find the derivative of $\frac{d}{dx}(\sin^{-1}x)=\frac{1}{\sqrt{1-x^2}}$ using implicit differentiation. The question is why do I need to use implicit differentiation when I have a function defined only in terms of $x$.
I'm trying to find the derivative of $\sin^{-1}(x)$. I know the steps that lead to $\frac{1}{\sqrt{1-x^2}}$, however I don't understand why the following reasoning leads to a wrong answer.  Because  $$\frac{d}{dx}f^{-1}(x) = \frac{1}{f'(f^{-1}(x))} $$  If we plug in for $f(x) = \sin(x)$, and because $\frac{d}{dx}\sin(x) = \cos(x)$ we get  $$\frac{d}{dx}\sin^{-1}(x) = \frac{1}{\cos(\sin^{-1}(x))} $$  Since $\sin(x) = \cos(x-\frac{\pi}{2})$, we can state that $\sin^{-1}(x) = \cos^{-1}(x-\frac{\pi}{2})$. (I suspect this is what is wrong)  Thus,  $$\frac{d}{dx}\sin^{-1}(x) = \frac{1}{\cos(\cos^{-1}(x-\frac{\pi}{2}))}$$  Then, by the definition of inverse function, we have  $$\frac{d}{dx}\sin^{-1}(x) = \frac{1}{x - \frac{\pi}{2}}$$
Editing Photoshop (PSD) files on Ubuntu? I was wondering that is it possible to edit PSD files on Ubuntu (like with Gimp). Actually, I need to edit layers. I haven't managed to do it. It would be nice if there is a chance! :) Why does US Robotics destroy Lanning's house? Shortly after the death of Dr. Lanning, Detective Spooner travels to the doctor's house to look for clues.  While he is searching the house a demolition bot (which had been programmed to demolish the house the next day) goes on a rampage and destroys the house with Detective Spooner in it.   The house is fully furnished and appears to be in good condition. Why would US Robotics want to destroy the house with everything in it? Why not sell the house and everything inside? Where can I find a reference to the format of the comment field (field 5) of the /etc/passwd file? In the /etc/passwd file on my system, the "comment" field, field 5, is inconsistent in its contents.  I thought that I could extract it to get the full name of the user.  fullname=`awk -F: '$1 == name {print $5}' name=$LOGNAME /etc/passwd`   However this returns with $fullname containing a name with 0, 3, or 4 commas following.  Exploring the man page (man 5 passwd) provides no details of this field other than describing it as "user name or comment field."  Perhaps there is additional information that is stored along with the user name? Unexpected workspace change? My friend who has only recently downloaded blender has come across a workspace issue. He said he accidentally pressed a button and it has entirely changed his user interface/ Workspace. He can't open, save or do anything with the change.    This is a full window image! How to find limit as $x \to 0$ of $x\sin(1/x)$?    For $x^2\sin(1/x)$, I know it's $0$ since by the Squeeze theorem, $-x^2 \le x^2\sin(1/x) \le x^2$, but for $x\sin(1/x)$, I run into some problems when applying Squeeze theorem. Why is the derivative of a vector orthogonal to the vector itself? $R(t) \cdot R'(t) = 0$, which is what every source I can find tells me. Even though I understand the proof I don't understand the underlying concept. If $R(t)\cdot R'(t) = 0$, then $R'(t)$ is orthogonal to $R(t)$, right?   But you use the same derivative to find the tangent of a curve. Then somehow if you differentiate the tangent itself, you get the normal to the curve.  I really can't wrap my head around this. Could someone help me understand? Following , I would like to know what exactly it means for a tag to be marked as blocked (formerly blacklisted) on a Stack Exchange site.  Does it mean I can't use that tag on my question? What are the reasons for blocking a tag? Who can block tags? Why am I being prompted to remove a tag from a question I'm editing? In the case of a non-inverting amplifier, if I DC couple the input signal with a capacitor only, \$C_1\$ (i.e. no resistor to ground), sources (as well as an experiment in the lab) show me that the op amp input will be saturated as the input bias current deposits charges on the terminal without having a return path to ground. This is solved by using connecting a resistor (\$R_1\$, as shown in the image below) to ground to create the return path. [The image is reproduced from an answer by "Neil_UK" on one of my previous questions, thanks Neil!]    My questions:   1) How is the input bias current able to continuously deposit charges and saturate the input if I don't use the resistor? The resistor should not allow the DC bias current through, so where is this current flowing from/into?  2) Why is this not a problem for the feedback path to the "-" terminal? There is no resistor to ground in that case, and capacitor \$C_2\$ seems to be causing no issues. The PDFs I've been reading suggest that the feedback behaves similar to when I have a resistor to ground by acting as the return path, how is this? How could I write a command in LaTeX, so after compiling it will be shown as a text in PDF and not executed?  For Example, I would write \it This is how you write in italic and the compiled pdf would contain the exact same text between the quotation marks without \it being executed. How can I find my Apple ID on my iPad? My mum can't find her apple ID on her iPad. I'm not with her at the moment and do not have an iPad so I don't know where to look.  She already has an Apple ID set up on this iPad, so I don't want to set up another one. If there was a black hole on earth, what would it look like? If an answer does exist, I'd love to hear it. I'm trying to incorporate a doomed earth story in something I'm writing, and the end of the world I'm going for is a black hole.  Let's say the black hole was created in the Philippines (I chose it because it's on the equator). Here's  for reference.  If I was standing 90 degrees of that, in Kenya, or if I was standing 180 degrees of that, in Mexico; what would I see in those two separate instances? What if I was standing in Japan above it, or in Australia below it?  Let's also say I was riding a boat towards the source of the black hole. Unlike most imagined scenarios where I am orbiting the black hole and slowly falling into it, this time I'm no longer orbiting it - I'm getting sucked into it instead. I'm imagining that the black hole will "half" be inside Earth and "half" will be in the atmosphere. But as it pulls me in, what will it look like to me (if I was massless, so no tidal forces)? It's probably not going to be a black hemisphere sticking out of the ground. Light will be lensed around it in some way so as to appear like a hole in the air.  Also, if the black hole was very small, it probably won't have a strong gravitational pull, and assuming it doesn't disappear, you'd be able to look at it (or at least, its effect on the light in its surroundings). Of course, as you approach it, it would still have an event horizon, so spacetime would still be warped if you're near enough and the surroundings will still experience tidal forces.  As the black hole gains mass, it will get larger, and so its event horizon will increase. If I was standing in Kenya and the event horizon "washes over me," (as in a water wave) what will I see?  I've so far imagined that, at least when seen from outer space, there will be a hole on the ground and light will be lensed around it. When you are on the ground, though, it becomes a bit harder to imagine.  I do hope someone can lend me a hand!  Edit: Someone had previously answered this question, but I'm not sure why it's been deleted. It was pointed out to me, though, that a black hole cannot stay in place. So an additional question for me is, does that mean it is impossible to be standing still, and the event horizon "washes over you"?  Edit2: Here's a guiding thought experiment that could answer the question. Imagine that you are in outer space directly above the black hole, looking at the black hole that is on the earth's surface. Can you imagine what it would look like? It would simply be a hole with the surrounding light gravitationally lensed around it (possibly lensing the entire earth if it's big enough - but let's say it's not).  Now imagine that you are on the direct opposite of where you were above - you're right behind the black hole this time, looking directly at the black hole but unable to see it because the earth is blocking your view.  Now try to imagine, from the second image, rotate the earth into the first image. I find it a bit hard to do. Just as the black hole is appearing as you rotate the earth, how would the light be lensed? In an alternating current, do electrons flow from the source to the device? If electrons in an alternating current periodically reverse their direction, do they really flow? Won't they always come back to the same position? Resize screen for Dead touch zone I have Star N7189 (5.5") with the lower zone of touch screen dead     The menu and back buttons don't work,but the physical home button works.   I tried to press hard on the edges of the screen, I tried to apply a twist from the short sides as if it were nothing but a wet towel. I did the hard reset but it did not work.  Sometimes pressing dead area, the phone takes it as menu button  As well as replacing the screen what other methods can I try? Is there any app, launcher or trick to resize the screen usable only to the working area?  P.S. a few hours after leaving it in airplane mode suddenly the touch had this problem but I do not remember if I inadvertently placed something heavy in that area of ​​the cell. Should I install programs from a source tarball (`.tar.gz`), from the Ubuntu Software Centre, or from elsewhere? Using ArcGIS 10.0, is it possible to reorder the fields in a file geodatabase, and have the order persisted to the database?  We considered exporting to XML, and writing a parser to change the field order, then re-creating a new database from the XML. This seems like a laborious process.  Another idea is to import the existing field definitionss into Excel, re-order them, export to CSV, then write a Python script to create these fields in a new Geodatabase. Does a script like this exist, before we write it from scratch?  Unfortunately the ArcGIS Diagrammer doesn't seem to support field re-ordering.  Are there any scripts or hacks we could use to simplify this process? Bash while read loop breaking early I've been banging my head against for wall for a while with this one.  I want to SSH into a set of machines and check whether they are available (accepting connections and not being used). I have created a small script, tssh, which does just that:  #!/bin/bash  host=$1 timeout=${2:-1}  ssh -qo "ConnectTimeout $timeout" $host "[ \`who | cut -f1 | wc -l \` -eq 0 ] &amp;&amp; exit 0 || exit 1"   This script works correctly. Returning 255 if there was a connection problem, 1 if the machine is busy and 0 if everything is good. If anyone knows a better way to do this please let me know.  So next I try and call tssh on my set of machines using a while read loop, and this is where it all goes wrong. The loop exits as soon as tssh returns 0 and never completes the full set.   while read nu ; do tssh "MYBOXES$nu" ; done &lt; &lt;(ruby -e '(0..20).each { |i| puts i }')   At first I thought this was a subshell problem but apparently not. Any help, along with comments on style/content, would be much appreciated! I know I'm going to kick myself when I find out why... I created a new profile for gnome-terminal and I can switch between &quot;Implicit&quot; profile and the new created profile as you can see in the below image:  Now I wonder how can I switch between profiles using a command-line/script or maybe using a shortcut for this.  Note: Solutions like: gnome-terminal --tab-with-profile=Implicit &amp;&amp; exit   are excluded because I prefer not to close and open another terminal or get another terminal window in this process of changing the profile. How to add a confirm dialog to a command button? How to resolve a Java Rounding Double issue I have asked this question on SO but perhaps that was the wrong location. Apologies for double posting but the question may have a better audience here.  The original question is:   And the text is below. Thanks.    When running the postgresql [9.3] "COPY TO" functionality, the postgres user creates a file. I have told it to put the file in the tmp directory (although this can be changed if needed).  This is run as part of a cron job, the user is an "export" user set up especially for this task. The end goal is for the exported file to end up in /home/export/created_file.csv.  1) postgres can't stash the exported file straight into the home directory of another user without setting up the folder to be overly permissive.  2) the export user can copy the file from /tmp to it's home directory, but then can not clean up (delete) the postgres owned file in /tmp.  This is a daily process and so old files will need to be deleted in some way. I can add a cron job to clean up the files afterwards but I was hoping for a cleaner solution - either a way to change the permissions on the file that postgres creates, or some way to give the export user the additional permission of being able to delete that file in advance of it's creation.  Giving the export user the same permissions as postgres in general however isn't an option, as it would then have free reign over the database.  Any advice, be it unix based or postgres based would be very welcome, thanks.