By the inverse function theorem, we know that $G'(x)=1/F'(G(x))$, where $G=F^{-1}$. I want to obtain $G''(x)$, but I don't know how to get the derivative of $F'(G(x))$. Any hints?
I know that the derivative of the inverse function of $f(x)$ is $g'(y) = \frac{1}{f'(x)}$ But how to derive the formula for the second derivative of g(y) knowing that $\left[\frac{1}{f(x)}\right]' = -\frac{f'(x)}{(f(x))^2}$ ?  I just started studying this chapter, so please try to be as simple as possible ;-) Thank you.
How to move a database from SQL Server 2012 to SQL Server 2005 What are my options if I need to move a database from SQL Server 2012(32bit) to SQL Server 2005(64bit)?  I know I cannot:   restore a backup of the database on SQL Server 2005 detach &amp; attach   I know I can:   use import data wizard, and I tried it on one database, but it only moves data, and even that was troublesome as I needed to do a lot of work creating temp tables to maintain identity columns, recreating all FKs, indexes etc.    Is there an easier option? pgfplots: prevent single plot from being listed in legend In order to fill the area between two complex curves I use tables and the vertcat command (see ).   Now I don't want the plotted area to appear in my legend, just the two curves. Is there any command to prevent the yellow area plot from being added to my legend?  this could be a minimal example:  \documentclass{article} \usepackage{pgfplots}  \begin{document} \begin{tikzpicture}   \begin{axis}[area style,axis on top]     \addplot+[mark=none,fill=yellow,draw=none] {0.1*x^2} \closedcycle;     \addplot[mark=none,draw=red,line legend] {0.1*x^2};     \addplot[mark=none,draw=blue,line legend] {0.0*x^2};     \legend{red curve,blue curve};     % \legend{\empty,red curve,blue curve};   \end{axis} \end{tikzpicture} \end{document}   With this legend{...}-command the last plot is skipped, an additional \empty doesn't help, too.  Sadly, drawing the first to plots together with fill=yellow,draw=red is not an option for me. Does anybody know any inside hacks? As far as I know, there is no command shown in the manual. Thanks in advance. For $x$ and $\lambda$ complex, if $|x| \gt \frac{1}{2} + \sqrt{\frac{5}{4} + |\lambda|}$ then $|x^2+\lambda| \gt |x| +1$ How to show that $2730\mid n^{13}-n\;\;\forall n\in\mathbb{N}$ Show that $2730\mid n^{13}-n,\;\;\forall n\in\mathbb{N}$   I tried, $2730=13\cdot5\cdot7\cdot3\cdot2$  We have $13\mid n^{13}-n$, by Fermat's Little Theorem.  We have $2\mid n^{13}-n$, by if $n$ even then $n^{13}-n$ too is even; if $n$ is odd $n^{13}-n$ is odd.  And the numbers $5$ and $7$, how to proceed? Macros for code annotations On OpenBSD and with OpenBSD's  and , is it possible to store the  of a user in a  repository of the same user?  What would be the proper way to accomplish something like this?  (To steer the answer into the correct direction, I would not be opposed to changing some permissions in the system, although I'd rather not have to recompile the binaries, nor violate any good security paradigms.) I need assistance in integrating $ \frac{x \sin x}{1+(\cos x)^2}$ Import/export 3ds format in Blender 2.8 I need to import and specially export in 3DS format. I know that I can import a file in 2.79 and the copy-paste to 2.8, but then when I copy-paste it back to 2.79 I'm having problems with textures missing. Is there any way to import/export 3ds in 2.8 without using any other version or software like we did in 2.79? The extension is missing in the add-ons repository either. Why does casting an ArrayList 's generic to a superclass not work? Which locations are traditionally Santa's home - i.e. where can you visit Santa Claus year-round? Obviously around this time of the year, Santa and his minions will be out and about, finding out who is naughty and nice, and delivering presents.  However, the rest of the year he's back at his home.  Where is that generally considered to be visitable?  Yes, in some traditions it's the 'North Pole', but I know at least one other country (Finland) has Rovaniemi as his home. Should a button become lighter or darker on hover? We're having a discussion in the office about whether a button should become lighter or darker when a user hovers over it.   Here are some examples from the field:  Apple "Buy Now" button (Second is hover, third is depressed) -     Twitter Bootstrap -    (Unselected)   (Hover)  Github homepage - github.com      The button on FogBugz homepage goes from yellow to slightly lighter yellow. The buttons on Optimizely and Visual Website Optimizer hardly change.  Amazon's "Buy Now" button doesn't do anything when you hover over it (besides change to a pointer cursor). The colored buttons in Google's new interface (see Gmail or Calendar) go slightly darker when you hover over them.  Finally here is our button:    Should your button become lighter or darker when you hover over it? What else should you consider? Does anyone have data on whether the hover effect matters for conversions? When is the completion of a ring a local ring ? Let $R$ be a commutative ring with unit and let $m$ be a maximal ideal of $R$. Are there known conditions on $R$ or $m$ such that the $m$-adic completion $\hat{R}$ of $R$ is a local ring.   Since the completion of a Noetherian local ring is again local, I'm primarily interested in cases where $R$ itself is not local.   An example is the polynomial ring $R=k[X_1,...,X_n]$ ($k$ a field) with $m=(X_1,...,X_n)$ where the power series ring $\hat{R}=k[[X_1,...,X_n]]$ is local with maximal ideal $(X_1,...,X_n)$. Fake small caps with XeTeX/fontspec? Using XeTeX and fontspec, I use the Liberation font family in of my documents. Unfortunately those fonts (esp. Liberation Serif) do not natively support small caps.  Is there a way to enable fake small caps in fontspec? Rotating an object continously with OnMouseDown I have a script that works but it does not do what I want it to do. My script makes an object rotate on when another is clicked. However, I want the object to continuously rotate until I click on the other object again. Right now when I click it rotates just a bit and stops.  Here is my script:   public class Rotate : MonoBehaviour {     public float speed =50f;     public GameObject target;      public void OnMouseDown(){        target.transform.Rotate (new Vector3(0, Time.deltaTime, 0) * speed);     }  }   How can I make the object rotate continuously until I click it again? Prove that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$ This is Problem 6 of the 2007 Indian National Math Olympiad (INMO).  If $x, y, z$ are positive real numbers, prove that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2).$  My best idea was to expand this and simplify. Although that doesn't look very feasible. Another idea is to see that $x^2+y^2+xy \geq x^2+y^2$. Then we just have to show that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(x^2+y^2)(x^2+z^2)(y^2+z^2)$ if that is even true. Subgroups of $U(n)$ isomorphic to a direct sum of cyclic groups Let $U(n)$ to be the set of all positive integers less then $n$ and relatively prime to $n$. Then $U(n)$ is a group under multiplication modulo $n$.  A. Find integer $n$ such that $U(n)$ contains subgroup isomorphic to $Z_5\oplus Z_5$.  B. Show that there is an $U(n)$ containing a subgroup isomorphic to $Z_3 \oplus  Z_3$, but not isomorphic to $Z_4 \oplus  Z_4$. We had a small widget that allowed customers to share records that were set to a private sharing model in our community. Everything was rather simple and it would simply allow them to select users that were visible via Sharing Sets and create an __Share record in APEX.  I've been reported recently that an error is thrown but only on the community side of things. Nothing seems to be working where it was working. We are using    Customer Community Plus licenses.  The error being thrown is : System.TypeException: DML operation INSERT not allowed on &lt;Object&gt;__Share which seemed to indicate access not available on the object. That seems odd because they are able to interact with the object just fine in order to create/edit/delete them.  I looked up the Sharing Model (its private both internally and externally), I looked at the profile and their access and the sharing sets in the community. Like I said, they seem to interact fine with the objects when it comes to seeing the records and their right.  Additionally, standard license users can use the component fine and also share with community users without a problem. They also see the records as expected.  Am I missing an update or something that seem to make it so this doesn't work anymore? Anyone having trouble with this or encountered the same problem? Can we add a section in our vf page and add components to it Can we add a section in our vf page and add components to it. I mean split my vf page into two parts and then add components to it. Best practice: card expiry date on checkout What is the best format for gathering the card expiry data during the checkout process on an eCommerce website?  So we need to collect the month of expiry and the date of expiry (as standard on all web card transactions).  I have seen this done differently on websites so it would be good to know the best practice. How do I calculate the CRC32 (Cyclic Redundancy Checksum) of a string in .NET?