The question I was solving was to find the derivative of $y = \arcsin(2x\sqrt{1-x^2}), -\frac{1}{\sqrt{2} } &lt;x&lt; \frac{1}{\sqrt{2}}$ Substituting $x = \cos\theta \Rightarrow \theta = \arccos (x)$, I got $y = \arcsin(\sin2\theta) = 2\theta$ Differentiating, I got $\frac{-2}{\sqrt{1-x^2}}$ However, substituting $x=\sin\theta$, simplifying and then differentiating, I got $y' = \frac{2}{\sqrt{1-x^2}}$, which is the negative of what I'd gotten earlier. Also, this is the correct answer according to my textbook. Both substitutions lead to $2x\sqrt{1-x^2}$ turning into $\sin2\theta$, so why is the substitution $x = \sin\theta $ more valid than the substitution $x = \cos\theta$?
In a problem, we have to differentiate : $$ y = \arcsin (2 x \sqrt{1 - x^2}),$$ given: $$-{1/(\sqrt2)} &lt; x &lt;{1/(\sqrt2)}$$.   My approach :  Put $x = \cos (p)$, so y reduces to:- $y = 2p = 2 \arccos (x)  \implies$ ${dy}/{dx}$ $ = -{2}/{\sqrt{1-x^2}}$   But the solution given is without the minus sign, so I am doing something wrong here but don't know what.  So please tell me what I am doing wrong and how to correct it.
Is it acceptable to take photos of slides at a conferences for private use? Is it accepted/appropriate to take pictures of presentation slides at conferences for private use?   I am not talking about putting those pictures out in the public afterwards, I know this is illegal without consent of the presenter.    Should it matter, I am talking about math conferences.  And I am asking because I don't see anyone taking pictures. How can I disable the Document Library drag and drop function? Before I start - I'm new to this SharePoint development lark and am slowly familiarising myself with it.  We're using SharePoint online and I have amended the SharePoint out-of-the-box document library to include some extra fields that are required when the user adds a new document.  If the user creates or uploads a new document then they are prompted to enter the required information. However if they use the drag and drop functionality the entry of the necessary fields is bypassed.  During my search I came across  which asks how to enforce metadata entry (which would be another good option for me). It also mentions disabling the drag and drop functionality by editing dragdrop.js and removing content from various methods.  I've had a look at the site collection I'm working on (to which I have full control) and I can't seem to find dragdrop.js (or the layouts folder for that matter).   Is this something that's held at site collection level or is it held at a higher level? I don't have admin rights above my site collection, do I need them? Do I need to use Visual Studio to make the necessary changes? (Currently I'm doing things through the web and having a bit of a play with Napa I don't have Visual Studio installed)  When I get there is it obvious what methods need to be removed? How can I tell SharePoint to use the new version of dragdrop.js? Is it just a case of saving a copy of the existing one to a different name and then editing the existing file?      I suppose the best place for me to have asked this question would probably have been on the existing post, but I am unable to post a comment on there, sorry... Where Android apps store data? How do I tunnel and browse the server webpage on my laptop? I run a web app in one of my lab servers, and I have already setup X11 Forwarding on those machines. Other lab members can tunnel through SSH, and browse that web app on their local browser at home.  I can't. Last time I checked with them, there is almost nothing I need to do.  When I type 192.168.1.113/webapp I get nothing.   Any tips?  THanks.    I ended up using the second method :)  ssh -L 8080:&lt;server-ip-address&gt;:80 &lt;username&gt;@&lt;remote-addr&gt; -N Prove that $f(x)\in \mathbb{Z}[x]$ such that $f(0)$ and $f(1)$ are odd has no integer roots Should I always use the perfect present, when the sentence uses "never"? In my English classes, I was taught that never should not be used with past forms, only with present perfect forms; this is also what I read in a grammar book I bought some years ago. (I bought the book in the USA; I guess the book is about American English grammar, but I don't have any reason to believe there is any difference between American, and British English in this case.)  Is it correct to use never with the past tense?     I never said that.        I have never said that. Exploiting JavaScript's eval() method Prove that $\frac{(n-2)s^2}{\sigma^2}\sim \chi^{2}_{n-2}$ Why exactly can't a single resistor be used for many parallel LEDs? Why can't you use a single resistor for a number of LEDs in parallel instead of one each? Isomorphism $H_n(D^n, S^{n - 1};A)$ and $A$ The homomorphism $a \mapsto [[a \cdot f]]$ is supposed to be an isomorphism between $A$ and $H_n(D^n, S^{n - 1};A)$ for any homeomorphism $f: \Delta^n \to D^n$. 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I am in the latest snapshot (16w03a). please help me (the person who answers my question will be given credit for it in my map) How to call C++ function from C? Recursively delete empty directories in Windows I have a directory on my Windows 7 machine that has hundreds if not thousands of sub-directories.  Some of them have files, some do not.  I want to delete all the empty directories.  Looking at the del and rmdir DOS command, it does not look like you can recursively do this without deleting all the files.  Is there a way to do this from the command line? From this , I now understand the gradient to be the directional derivative that returns the steepest slope at a point. However, reading my textbooks, they all say that the gradient is normal to the tangent plane. In the above SE question, the gradient sounds like it should be parallel to a direction in the $xy$ plane, not actually up on the surface.   My question is therefore do I have a misunderstanding of the tangent plane, gradient or directional derivative?   In addition, I question how the gradient always passes through the origin or the $xy$ plane? If for instance in the graphic, the gradient had been another direction (say horizontal), then it wouldn't pass through the origin! In addition, the gradient of other points in the graphic would satisfy this example. Many thanks! Approximation of measurable function by simple functions Force ssh to not print "Remote host identification has changed" warning Is there a way to avoid ssh printing warning messages like this?   @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ @    WARNING: REMOTE HOST IDENTIFICATION HAS CHANGED!     @ @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@   Although the remote host identity has changed but I know it is fine and  just want to get rid of this warning. Quoting columns with spaces in PostgreSQL? I migrated a database from SQL Server to PostgreSQL.  Most column names contain double words, for example:  SELECT [Column Name] FROM table;   ...which does not work in PostgreSQL.  What is the correct syntax for PostgreSQL? The app forgets that I voted when I scroll the question Related:   I am using 1.0.71 on KitKat.  Immediately after voting, the arrow is displayed in red and the vote count is correct. However, when I scroll to read answers and then back to the top, the arrow is grey again and the vote count is reset to the previous value.  A reload of the question (pulling the finger down) is needed to show the correct situation, which is kept from then on.  This case was on security.stackexchange.com but I observed the same behavior on other sites too.  Screenshots   original state:  upvoted:  scrolled down and back up  after a reload Movie with "Where do the tubes go?" I'm trying to find a movie with an underground facility with people wearing white suits. The people have been affected such that they show no curiosity. There is one person that does and asks one of his co-workers where a tube leads to.      At the end there is an uprising where everyone gets to see the surface and sunlight.   What is the name of this movie? In Minecraft, is there any way to make tools indestructible (without using mods)? I want to make some tools indestructible (for an adventure map). Is it possible to do this using enchantments (or a command of some kind?)