What is derivative of $\sin\sqrt x$ using the definition of the derivative ?  The definition of derivative is:-  $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$
The definition of derivative of a function $f(x)$ is $$\lim_{h\to0} \frac{f(x+h)-f(x)}{h}$$  Using this definition, the derivative of $\sin\sqrt{x}$ will be:  $$\lim_{h\to0} \frac{\sin\sqrt{x+h}-\sin\sqrt{x}}{h}$$  $$\lim_{h\to 0} \frac{\cos\left(\frac{\sqrt{x+h}+\sqrt{x}}{2}\right)\sin\left(\frac{\sqrt{x+h}-\sqrt{x}}{2}\right)}{h}$$  Now i got stuck. How to find the limit or simplify this expression?  I get intuition that we have to use $$\lim_{x\to0}\frac{\sin x}{x} = 1$$  but that too is leading no where. I am unable to remove h from denominator.  NOTE  I know the derivative of $\sin\sqrt{x}$ is $\frac{\cos\sqrt{x}}{2\sqrt{x}}$ using chain rule, but this exercise was given to us for practice using division quotient.
How do I install minecraft 1.9 on mac? How to get a cross-origin resource sharing (CORS) post request working Does a logistic regression maximizing likelihood necessarily also maximize AUC over linear models? Gnome Panel becomes fully transparent when window is not maximised! Create a table with LaTeX PSTricks error "Undefined control sequence  \c@lor@to@ps" I have a problem with this file tex:  \begin{document} \begin{figure} \begin{center} \begin{pspicture}(0,0)(1,3.8) \psline{-&lt;}(0,0)(0,.2) \psline{-}(0,.2)(0,.4) \psline{*-&gt;}(0,.4)(0,.9) \psline{-*}(0,.9)(0,1.4) \psline{-&lt;}(0,1.4)(0,2.4) \psline{-}(0,2.4)(0,3.4) \psline{*-}(0,3.4)(0,3.6) \psline{&gt;-}(0,3.6)(0,3.8) \rput(1,.4){$y=0$}%testo \rput(1,1.4){$y=1$} \rput(1,3.4){$y=3$} \end{pspicture} \end{center} \caption{} \end{figure} \end{document}   but I don't understand where is the error. use contradiction to prove that the square root of $p$ is irrational Total width of element (including padding and border) in jQuery As in the subject, how can one get the total width of an element, including its border and padding, using jQuery?  I've got the jQuery dimensions plugin, and running .width() on my 760px-wide, 10px padding DIV returns 760.  Perhaps I'm doing something wrong, but if my element manifests itself as 780 pixels wide and Firebug tells me that there's 10px padding on it, but calling .width() only gives 760, I'd be hard pressed to see how.  Thanks for any suggestions. Question:      Let $p$ be any prime and $n \geq 3$. Show that there exists a non-abelian group of order $p^n$.    Attempt:   Take $n = 3$. Writing $\mathbb Z_p \times \mathbb Z_p = \{e, \alpha_1, \ldots, \alpha_{p^2 - 1}\}$ and considering $$\begin{align}\tau_{jk} : \mathbb Z_p \times \mathbb Z_p&amp;\to \mathrm {Aut} (\mathbb Z_p)\,\\e &amp;\mapsto id\\\alpha_j &amp;\mapsto id \\\alpha_i &amp;\mapsto \rho_k  \end{align}$$  $\forall i \neq j$ where $\rho_k : \mathbb Z_p \to \mathbb Z_p$ is such that $\rho_k (1) = k$, with $k \in \{2, \ldots, p-1\}$. Then we take the group $(\mathbb Z_p \times \mathbb Z_p) \ltimes_{\tau_{jk}} \mathbb Z_p$.   Then $$(\alpha_i , 2) \ltimes_{\tau_{jk}} (\alpha_i, 1) = (2\alpha_i, 1  + \tau_{jk} (\alpha_i)(2)) = (2\alpha_i , 1  +2 k)$$  and $$(\alpha_i , 1) \ltimes_{\tau_{jk}} (\alpha_i, 2) = (2\alpha_i, 1  + \tau_{jk} (\alpha_i)(2)) = (2\alpha_i , 1  +k)$$  taking $k = 1$ for example we have that the group $(\mathbb Z_p \times \mathbb Z_p) \ltimes_{\tau_{jk}} \mathbb Z_p$ is not abelian.    I was wondering if this is a good aproach. If not, is there an easier way to reach the result?  If this is the "best" way then would the higher cases ($n &gt; 3$) be more of a notation play game? A question on $p$-groups, and order of its commutator subgroup. $\textbf{QUESTION-}$ Let $P$ be a p-group with $|P:Z(P)|\leq p^n$. Show that $|P'| \leq p^{n(n-1)/2}$.  If $P=Z(P)$ it is true. Now let $n &gt; 1$, then  If I see $P$ as a nilpotent group and construct its upper central series, it will end , so let it be,  $e=Z_0&lt;Z_1&lt;Z_2&lt;......&lt;Z_r=P$   Now as $Z_{i+1}/Z_i=Z(P/Z_i)$, so if if I take some $x\in Z_2$\ $Z_1$ then $N$={$[x,y]|y\in P$} $\leq Z_1(P)$ and $N \triangleleft P $, so $P/N$ is a group with order $\leq p^{n-1}$.  Now if I let $H=P/N$ then obviously |$H/Z(H)$|$\leq p^{n-1}$.  Now $H'\cong P'N/N \cong P'/(P' \cap N)$ so from here I could finally bring $P'$ atleast into the picture, now |$P'$|=$|H'|.|P'\cap N|$ so $|P'|\leq |H'||N|$.   This is where I am $\textbf{STUCK}$  Now , from here how can I calculate or find some power $p$ bounds on $|H'|$ and $|N|$ so i could get my result. If $f:\mathbb{R}\to\mathbb{R}$ is continuous and $$\lim_{x\to\infty}f(x)=\lim_{x\to-\infty}f(x)=\infty$$  Prove that $f$ has a minimum on $\mathbb{R}$; that is, there is an $x_m\in\mathbb{R}$ such that $$f(x_m)=\inf_{x\in\mathbb{R}}f(x)&lt;\infty$$  I think the best approach would be to prove the existence of a lower bound, that is $\exists M\in\mathbb{R}:M&lt;f(x),\forall x\in\mathbb{R}$, but I don't know how to do this. I am probably supposed to use the extreme value theorem or the intermediate value theorem, but I don't know how to do that. Render scene with background jpg in python Fairly new to python and blender, so apologies if this is super basic. I've searched around and haven't found an answer.  I'd like to render some procedurally generated objects with a background image. In case it makes a difference, I'm using:    I don't need any interaction between the generated object and the background image so I've left a skeleton script I'm using to render below  import os DIR ="/home/path/to/folder" bg_file = os.path.join(DIR, "myimage.jpg") bg_img = bpy.data.images.load(bg_file) bpy.data.scenes['Scene'].render.filepath = os.path.join(DIR, "image") bpy.context.scene.camera = bpy.data.objects['Camera'] bpy.ops.render.render(write_still=True)   I managed to find some scripts that seem to be trying to do what I want online, but none that I could get to work. e.g.  import bpy filepath = "/home/path/to/folder" img = bpy.data.images.load(filepath)  for area in bpy.context.screen.areas:     if area.type == 'VIEW_3D':         space_data = area.spaces.active         bg = space_data.background_images.new()         bg.image = img         break  cam = bpy.data.cameras.new("Camera") cam_ob = bpy.data.objects.new("Camera", cam) bpy.context.scene.camera = cam_ob bpy.context.scene.objects.link(cam_ob) bpy.context.scene.objects.active = bpy.context.scene.objects["Camera"]   Any thoughts/tips would be greatly appreciated! Why my DAC works without GND and VCC? This is the most interesting thing I have seen! I have built a  that sends data (D0-D7) to a DAC (ADV2175). The design worked fine and I just have a low frequency alias noise (). This noise was present on all my ground planes (This is in prototyping stage and that's not unexpected from the test PCB).  I started tracing the noise path by disconnecting wires to see when the noise is gone. Surprisingly even after disconnecting all VCC and GND connections (+ all VREF and logics gone to 0 or 1) the noise was present yet. The more unexpected behavior is when the DAC is just connected to DATA port+ CLK pin (all coming from the source FPGA, NO VCC or GND), it continues working and a sine wave is produced (but with the same noise)!!!  First I thought it may receive its VCC/GND form the logic 1/0 fed into it but if it was correct, it should not have produce 0x00 and 0xFF levels, but it makes those levels correctly (I changed 0xFF data stage to another level and a notch appeared on top of the waves. this shows it can build oxFF correctly).    This is a big challenge for my PCB design, as if this device really takes its GND from the data port, instead of the dedicated pins, I will have a hard time for designing a reliable ground plane for it (the return voltage should go back to a logic noisy environment and analog-digital ground plane isolation may be impossible).  Can anyone explain what is happening there? Make file in linux readonly Speed of light travel Test for convergence of the series $\sum_{n=2}^\infty\frac{1}{(\ln n)^{\ln n}}$ Could I have a hint for testing the convergence of the following series please?  $$\sum_{n=2}^\infty\frac{1}{(\ln n)^{\ln n}}$$  Edit  The integral test does not work because $\int_1^n\frac{1}{(\ln x)^{\ln x}}dx$ has not an elementary primitive.  Thank You. Is it possible to set start-up applications to initialize in certain workspaces? I'd like to not have to manually distribute my start-up applications across my numerous workspaces every time I boot up my machine. I'm running Ubuntu 12.04 .  Any Help ? Looking for a post-apocalyptic trilogy where pilot is nursed back to health by natives of the land I read a series of books around 2005. I think it might have been a trilogy.  Basic story line is in a post apocalyptic world. The main character is a pilot for the Federation or something similar to that. He crash lands, is nursed back to health by the natives, comes to like and respect them, and does not want to fight against them.  Also in the later book I think it also involves another race who are very Japanese-ish. They drink sake and the "clan" names are things like Fu-Jitsu and Mitsu-Bishi. When applying dropout in artificial neural networks, one needs to compensate for the fact that at training time a portion of the neurons were deactivated. To do so, there exist two common strategies:   scaling the activation at test time inverting the dropout during the training phase   The two strategies are summarized in the  below, taken from .  Which strategy is preferable, and why?    Scaling the activation at test time:    Inverting the dropout during the training phase: I want to write a conditional expression such as the following.    What is the best way to express such conditional expressions in Latex?