It is well known that the formula $$\int x^n dx = \frac{x^{n+1}}{n+1}+C$$  breaks down at $n=-1$ (we get a logarithm).  My question is, does there exist a value of $n \in \mathbb{Z}$ for which the formula $$\frac{d}{dx}x^n = nx^{n-1}$$  breaks down?
Prove that $\frac d {dx} x^n=nx^{n-1}$ for all $n \in \mathbb R$.   I saw some proof of $\frac d {dx} x^n=nx^{n-1}$ using binomial theorem, which is only available for $n \in\mathbb N$. Do anyone have the proof of $\frac d {dx} x^n=nx^{n-1}$ for all real $n$? Thank you.
I want to subtract a constant number from my data to get new data file. For example, I want to subtract  10 from my data.   data.txt  Id  a   b   c   d   e 1   1   2   3   4   5 2   2   3   4   5   6 3   3   4   5   6   7 4   4   5   6   7   8 5   5   6   7   8   9 6   6   7   8   9   10 7   7   8   9   10  11 8   8   9   10  11  12 9   9   10  11  12  13 10  10  11  12  13  14 11  11  12  13  14  15 12  12  13  14  15  16 13  13  14  15  16  17 14  14  15  16  17  18   Required output:  out.txt   Id  a   b   c   d   e 1   -9  -8  -7  -6  -5 2   -8  -7  -6  -5  -4 3   -7  -6  -5  -4  -3 4   -6  -5  -4  -3  -2 5   -5  -4  -3  -2  -1 6   -4  -3  -2  -1  0 7   -3  -2  -1  0   1 8   -2  -1  0   1   2 9   -1  0   1   2   3 10  0   1   2   3   4 11  1   2   3   4   5 12  2   3   4   5   6 13  3   4   5   6   7 14  4   5   6   7   8 Jquery Validation for dynamically added rows that has multiple element with same name Photoshop CC 2019: Pixelated Brush Stroke For some reasons my brush stroke suddenly looks like this:     As the picture showed, the edges are jagged and pixelated. It doesn't happen before and it's just one of the built-in inking brush.  I know there are a lot of similar questions here but please don't mark this as duplicate. I double checked all the thing the other answers suggested: It's brush tool, not pencil, and my mode of brush and layer is Normal, not Dissolve. It's pixelated even if I draw with mouse so it's not tablet driver issue. I disabled GPU render but nothing changed. How do I fix this? On Takodana,      the main characters and various bar patrons can see the planet Hosnian Prime being destroyed by Starkiller Base with their (respective) naked eyes. How did they see this? Is the Hosnian system close enough to see? If it's not very close and they could just see the explosion, how did the light from it reach them so quickly? I have some non-Gaussian distributed variable and I need to check if there are significant differences between the values of this variable in 5 different groups.  I have performed Kruskal-Wallis one-way analysis of variance (which came up significant) and after that I had to check which groups were significantly different. Since the groups are kind of sorted (the values of the variable in the first group are supposed to be lower than the values of the variable in the second group which are supposed to be lower than the values of the variable in the third group and so on) I only performed 4 tests:  Group 1 vs Group 2 Group 2 vs Group 3 Group 3 vs Group 4 Group 4 vs Group 5   I have performed this analysis with two different methods. I started by using Dunn's Multiple Comparison Test but nothing came up significant. On the other hand if I use Mann-Whitney test and correct for the number of tests (4) using Bonferroni, 3 tests come up significant.  What does it mean? Which results should I trust? Making adult decisions "is" or "are" really not fun: Which is correct? Making adult decisions are/is really not fun.     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Does someone know a useful trick or item I could use? $$A=\begin{array}{ccc} 0 &amp; 1/2 &amp; 0 &amp; \cdots &amp; 0 \\ 1/2 &amp; 0 &amp; 1/2 &amp;\cdots&amp; 0\\ 0 &amp; 1/2 &amp; 0 &amp; \cdots &amp; 0\\ \vdots &amp; \vdots &amp; \vdots &amp;\ddots &amp; 1/2\\ 0 &amp; 0 &amp; 0 &amp; 1/2 &amp; 0 \end{array}$$  I need to know the spectral radius of $A$ in function of the dimension of the matrix $n$. I know only one way to know the spectral radius, using the definition I find all eigenvalues and then find the maximum. but it very hard if $n$ is changing. Some help please!  I did some iteration by   $$det(\lambda I-A)$$  $$n=2 \quad r=\frac{\sqrt{1}}{2}$$ $$n=3 \quad r=\frac{\sqrt{2}}{2}$$  for $n=2$ and $n=3$ are correct.  and if we take this rule, for   $$n=5\quad \quad r=\frac{\sqrt{4}}{2}$$  but this isn´t true since I know radius for this matrix should be $&lt; 1$ and if we take this rule  for $n=5$ we have  $$\frac{\sqrt4}{2}=1$$   Some hint? I am running on a dual core AMD E450 CPU. When I tried to run a 64-bit OS that requires hardware virtualization using VirtualBox, it showed me an error message that "AMD-V is not enabled".    My AMD processor is supposed provide AMD-V support, but I can find no option for enabling AMD-V in BIOS.    How can I solve this problem? How can I enable AMD-V for my CPU?  Thanks in advance  lscpu :-  Architecture:          x86_64 CPU op-mode(s):        32-bit, 64-bit Byte Order:            Little Endian CPU(s):                2 On-line CPU(s) list:   0,1 Thread(s) per core:    1 Core(s) per socket:    2 Socket(s):             1 NUMA node(s):          1 Vendor ID:             AuthenticAMD CPU family:            20 Model:                 2 Stepping:              0 CPU MHz:               1650.000 BogoMIPS:              3291.72 Virtualization:        AMD-V L1d cache:             32K L1i cache:             32K L2 cache:              512K NUMA node0 CPU(s):     0,1   EDITED:-  Error message from VirtualBox:-  Failed to open a session for the virtual machine XXX.  AMD-V is disabled in the BIOS. (VERR_SVM_DISABLED).  Result Code: NS_ERROR_FAILURE (0x80004005) Component: Console Interface: IConsole {1968b7d3-e3bf-4ceb-99e0-cb7c913317bb} Rejecting a PhD offer after having accepted it. Advice? $\operatorname{rank}(T^3) + \operatorname{rank}(T) \geq 2\operatorname{rank}(T^2)$ for finite-dimensional $V$ I looked up several tutorials on how to install and they dont work. I do everything correctly but then i get this:     The file   '/home/mike/Downloads/minecraft.jar'   is not marked as executable.  If this   was downloaded or copied from an   untrusted source, it may be dangerous   to run.  For more details, read about   the executable bit.   I run the minecraft.jar with the Sun java runtime 6 but it doesnt work:/. Any help would be greatly appreciated! Is the CMB rest frame special? Where does it come from? It seems that we are . This is the speed of our galaxy relative to the cosmic microwave background. Where does this rest frame come from? Is this special in any way (i.e., an absolute frame)? EDIT: I think the more important question is &quot;where does the CMB rest frame come from?&quot;. Inverse elements in the absence of identities/associativity. Where's the Rainbow Jetpack gone in Jetpack Joyride? Why did Zobrist leave the trail to the virus in the first place? What if I accidentally run command "chmod -R" on system directories (/, /etc, ...) I accidently ran  sudo chmod 755 -R /   instead of  sudo chmod 755 -R ./   I stopped it after few seconds, but now there is some problems such as  sudo: must be setuid root   How can I revert permissions back?