Proving the connection between the Binomial Theorem and the product rule for derivatives Let $a(x)$ and $b(x)$ be smooth functions, i.e they are infinitely times differentiable.  I have made the assumption that the derivative for the function   $$f(x)= (a\cdot b)(x)$$  can be given by   $$f^{(n)}(x)=\sum_{k=0}^{n}\binom{n}{k}a^{(k)}b^{(n-k)}(x)$$  Where I have defined $g^{(n)}$ as the $n-$ th derivative of some function.  I have made the observation when I was trying to calculate the derivative of such a function and I noticed that the derivatives are in the form:  $$f^{(1)}(x)=a'b+b'a$$  $$f^{(2)}(x)=ab''+2a'b'+a''b$$  $$f^{(3)}(x)=ab'''+3a'b''+3a''b'+a'''b$$  What I have tried so far is induction but I don't know how to manipulate the formula to get the result I want  $${f^{(n+1)}=f^{(n)}}^{'}=(\sum_{k=0}^{n}\binom{n}{k}a^{(k)}b^{(n-k)})^{'}=(\sum_{k=0}^{n}\binom{n}{k}[a^{(k+1)}b^{(n-k)}+a^{(k)}b^{(n-k+1)}])$$  Now I cannot say something like :  $$[\uparrow]=(\sum_{k=0}^{n}\binom{n}{k}[a^{(k)}b^{(n-k)}(a+b)]) $$   Because they are derivatives and they do not necessarily behave like exponentials. But I don't know if it even would help me  Because in the end I want to have a term that Looks like  $$\sum_{k=0}^{n+1}\binom{n+1}{k}a^kb^{n-k}$$  But I don't know how to do it.
How is Leibniz's rule for the derivative of a product related to the binomial formula? Possible Duplicate:          The  describes the expansion of the $n$th power of the sum $(a+b)$:  $$(a+b)^n = \sum_{k = 0}^n {n\choose k}a^kb^{n-k}$$  In calculus, there is a generalization of the product rule called , which describes the expansion of the $n$th derivative of the product of two functions $f$ and $g$:  $$(fg)^{(n)} = \sum_{k = 0}^n {n\choose k}f^{(k)}g^{(n-k)}$$  It's not difficult to prove using the product rule and induction, what I'm really trying understand here is why the formula is true. Is there a way to prove it using the binomial formula? How are these two formulas related? Is there a combinatorial proof for Leibniz's rule like there is for the ?   I understand this question is a bit vague, but I hope you'll agree that these two formulas look so similar it just feels like there should be some relation between the two. Of course this is not necessarily true, but I'm hoping it's not just a crazy coincidence.
80's Animated movie: An old man joins mission to save a boy stranded on a hostile world can anyone help me with this, I got stuck on it.  let $a_n$ and $b_n$ (for $n$ a non-negative integer) be two sequences of real numbers such that $\lim_{n\to \infty}a_n=a$ and $\lim_{n\to \infty}b_n=b$. prove that $\lim_{n\to \infty} \frac{a_0b_n+a_1b_{n-1}+....+a_nb_0}{n}=ab$. Are the complements of two homeomorphic compact, connected subsets of $\mathbb{R}^2$ homeomorphic? Proof: $\cos^p (\theta) \le \cos(p\theta)$ I came across this problem when I was at a book store inside of a book made to prepare Berkeley graduates to pass a mandatory exam. I wanted to buy the book, but, alas, I didn't have the money (forty bucks is a lot of money when you don't have a job). So I took my phone out and started taking as many pictures as I could. Unfortunately, I didn't take any pictures of the solutions!  Enough backstory. Time for math. The problem is as follows:  Prove that $\cos^p( \theta) \le \cos(p \theta)$ if $0\le\theta\le\frac\pi2$ and $0\le p\le 1$.  I tried using the series expansion for cosine, but that seemed to be a dead end. Then I tried using Euler's theorem, but I got stuck. Then I got distracted and started to think about other cosine identities. For example, $$\cos\left(\frac\theta2\right)=\pm\sqrt{\frac{1+\cos(\theta)}{2}}.$$ Then what is $\cos(\frac\theta3)$ equal to? I tried to figure it out then I realized that I needed to find the root of a cubic polynomial. Then I realized that I'm better at staying focused than finding the roots of cubic polynomials.  Anyways, a proof would be nice here. I really appreciate any hints or answers. I apologize for my digressions! Question: If $R$ is a finite boolean ring, then show $R \cong \mathbb Z_2 \times \mathbb Z_2 \times \cdots\times \mathbb Z_2$.   I know that   $\mathrm{char}(R) =2$  $R$ has $2^k$ elements for some $k$  (this is because, if $p\neq 2$ is a prime with $p\mid|R|$, then there exists a non-zero $x \in R$ s.t. $px=0$ and $p $ is odd $\implies x=0$, contradiction).   Does this suffice to answer my question? Why don't C++ compilers define operator== and operator!=? I am a big fan of letting the compiler do as much work for you as possible.  When writing a simple class the compiler can give you the following for 'free':   A default (empty) constructor A copy constructor A destructor An assignment operator (operator=)   But it cannot seem to give you any comparison operators - such as operator== or operator!=.  For example:  class foo { public:     std::string str_;     int n_; };  foo f1;        // Works foo f2(f1);    // Works foo f3; f3 = f2;       // Works  if (f3 == f2)  // Fails { }  if (f3 != f2)  // Fails { }   Is there a good reason for this?  Why would performing a member-by-member comparison be a problem?  Obviously if the class allocates memory then you'd want to be careful, but for a simple class surely the compiler could do this for you? Django template how to look up a dictionary value with a variable See code:   var file1 = "50.xsl"; var file2 = "30.doc"; getFileExtension(file1); //returns xsl getFileExtension(file2); //returns doc  function getFileExtension(filename) {     /*TODO*/ } Phone : nexus 5 version 2013 Android version : 6.0 Kernel version : 3.4.0-g2aa165e Build number : MRA58N   For the last few days, 10118 has been draining my phone's battery. It drains battery by using GPS. I typed 'ps' in a terminal emulator, and there was no application with PID 10118. I don't know what exactly 10118 is. My guess is that it's one of recently installed weather apps.  How can I fix this?    Screenshots (click images for larger variants) How to upload files to server using JSP/Servlet? Executing multiple commands from a Windows cmd script I'm trying to write a Windows cmd script to perform several tasks in series. However, it always stops after the first command in the script.  The command it stops after is a maven build (not sure if that's relevant).  How do I make it carry on and run each task in turn please?  Installing any software or configuring the registry etc is completely out of the question - it has to work on a vanilla Windows XP installation I'm afraid.  Ideally I'd like the script to abort if any of the commands failed, but that's a "nice to have", not essential.  Thanks. Sorry for the vague title--I don't want the title to be a spoiler.  I've seen the movie three times, and the last two times I looked specifically for clues to this question, but found none.  The final scene in  shows       Cobb's top spinning, as he goes off to be reunited with his children.  The top continues to spin, then the movie ends, without telling us if the top topples--and therefore revealing whether he's in the "real world" or still in a dream. Do we know if he was in a dream at the end of the movie?   I'm assuming the answer is that we simply cannot know.  But did I miss anything?  Were there clues that I missed? Delayed erasure experiment Why do you cast downvotes on answers? This is a question which started off with a , then turned into an , and now it's a question in its own right (as suggested by Jeff). I'm not sure where it could go from here :)  It's obvious that different people cast downvotes for different reasons. (Upvotes are less controversial, generally.) Here are some possible reasons:   Answer is wrong Answer is misleading (may be technically accurate, but will lead to readers making mistakes) Answer doesn't match the question (e.g. a C# answer to a question about Java) Answer doesn't really address the question, e.g. suggesting a completely different solution, even if the question specifies that there are good reasons why the current approach has to be followed Answer is okay, but not as good as another one which currently has fewer votes Answer appears to be plagiarised from existing answers. Answer is by someone I don't like. Answer is abusive (profanity, spam etc). No reason, I just felt like it. I'm a , and prefer down to up. Have you considered down? Ideological grounds: answer suggests a technology I don't like, or expresses a negative opinion of a technology I do like   Personally I go with 1, 2 and 8 - although I'll add comments suggesting changes in various other situations.  Have I missed anything? What do you do, and why? What are the net benefits (to you, the world in general, and the answerer) from your approach? My Sophix tablet was working without the SD card, but now the camera will not work. I get the following error message:     Insert a SD card before using. What is the exact difference between an autoregressive (AR) and vector autoregressive model (VAR)?  I always thought that VAR would just be for more than two variables, until I learned that AR can also have more than two variables. I.e. I had a misunderstanding(?) that AR models are not always bivariate.  Before I write too much potentially false assumptions about the difference between AR and VAR, it would rather be interested to see if anyone can make provide a quick and good summary. (Built-in) way in JavaScript to check if a string is a valid number In Lieb and Loss's Analysis, I saw that they mentioned  $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$ (dense wrt the $L^2$ norm, I think). But I didn't find its proof in the book. So I wonder why it is?    Does this conclusion hold for $L^1(\Omega, \mathcal{F}, \mu) \cap L^2(\Omega, \mathcal{F}, \mu)$ for any measure space $(\Omega, \mathcal{F}, \mu)$?  Are there similar statements if replace $L^1$ with $L^p$, and  $ L^2$ with $L^q$ for $p \leq q \in (0, \infty ]$ or $\in [1, \infty]$?  Thanks and regards! Can Excel show a formula and its result simultaneously? I want to return two objects from a Java method and was wondering what could be a good way of doing so?  The possible ways I can think of are: return a HashMap (since the two Objects are related) or return an ArrayList of Object objects.  To be more precise, the two objects I want to return are (a) List of objects and (b) comma separated names of the same.  I want to return these two Objects from one method because I dont want to iterate through the list of objects to get the comma separated names (which I can do in the same loop in this method).  Somehow, returning a HashMap does not look a very elegant way of doing so.