I know the limit definition of a derivative. And I read at the end that if the limit exist then we say that function is differentiable  and this existing limit is denoted by  $\dfrac{d}{dx}$ . so we say that $\dfrac{d}{dx}$ is a notation.  But during my graduation , in many proof of some theorems we do a step  as  Say.....  $\dfrac{d(f(x))}{dx} = g(x)$  Then multiply both sides by $dx$ we get... $d(f(x))= g(x)d(x)$  My question is that how we do this step....  Then i  read total derivative of a function and  it help me little bit .but don't understand completely ...... We used $\dfrac{d}{dx}$ as a notation  so how do above step .... Please help me to understand it....
In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $dy/dx$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula $dy = f'(x)dx$ we are able to plug in values for $dx$ and calculate a $dy$ (differential). Then if we rearrange we get $dy/dx$ which could be seen as a ratio.   I wonder if the author says this because $dx$ is an independent variable, and $dy$ is a dependent variable, for $dy/dx$ to be a ratio both variables need to be independent.. maybe?
I'm struggling to understand the difference between the standard error and the standard deviation. How are they different and why do you need to measure the standard error? How do real and effective user IDs work? When a normal user wants to make change to the passwd file, the user will by setuid be given the effective user access. User becomes root temporarily and can edit passwd.  However you can only edit your password right, and not everybody else? However your effective user access is root. So how come you're not allowed to change other passwords beside yours?  When you run a program with setuid, what does it mean actually when effective user is root, but real user id is still your name? Ubuntu 20.04 interface font too small after restart, even with high scaling factor Triangular Factorials I came across a statement online and have been looking for a proof :  It states that 1, 6 and 120 are the only numbers which are both triangular and factorials.  Is there any way I can prove this? This claim looks too 'big' and I've tried to prove it but  I couldn't. Can anyone help me to prove this? I am trying to use netcat to send a simple message over TCP, e.g. I want to send the digit 1.  So I understood that I write the following in the terminal, after installing netcat  netcat [ip-address] [port]  But how do I specify the message to be sent? i.e. where to I write "1" ? Does $P(A\cap B) + P(A\cap B^c) = P(A)$? Product of an exponential function of IDD variables if $x_i$ are IID  continuous random variables, with $E[x_i]=μ$, is the following correct?   $E\left[\prod_{i=1}^n(1+⍺\space x_i)^i\right] = \prod_{i=1}^nE\left[(1+⍺\space x_i)^i\right] =  \prod_{i=1}^n(1+⍺\space E\left[x_i\right])^i = (1+⍺\space μ)^n$  thanks! How does body fat leave your body? So I know that it’s not from tears or sweat because they are composed of mainly water and sodium, it’s not from going poo or pee as poo and pee are made of the parts of food that supply your body with no nutrients and your body also doesn’t “burn body fat” because there is a theory that no matter is ever destroyed so it’s none of those but another thing is it’s not your hair, ear wax, finger nails or anything of that sort because people can lose over 17 pounds in 3 months but you can’t build up that amount of hair , ear wax or whatever else in three months so how does body fat leave your body Has anyone had any success using twitters typeahead.js within visualforce pages? What is the difference between implication symbols: $\rightarrow$ and $\Rightarrow$? I have passed the calc sequence, and thought that I understood things.  But I have trouble with limits now and again.  From the way I understand it, the 'best' way to evaluate a limit, is to evaluate the expression at the number in the limit.  In that way it's not only a limit, but an algebraic expression (so, I think, the need for limits goes away entirely, superseded by accuracy.)  But then I see something like $\lim_{x\to 2} \frac{x^2-4}{x-2}$.  Now, I think in my mind 'the best way to evaluate this is just to plug in $2$'  But it doesn't work that way, as the limit is $4$, and if I plug in, the answer is $0/0$.  I think that people here, with their natural talent in math, can perceive where my train of thought is going wrong, and correct me.  Please! How do I test if all minecraft players on a server are in a radius? This has got me. I need a command that tells me if all players on a server are in a radius of a command block. If one of them isn't in the radius, the command will be false. Any ideas? Thanks! LaTeX Classes or Styles for Schedules and/or Calendars? Learning Regular Expressions Curious badge bug I saw that it has been already reported that also they have earned courious badge but the badge has not been given. but their answers say that they must wait a day and the badge will be awarded.  In my case its 8/5 and it was 5/5 long time ago and I have never received a message telling that I have earned it. its just telling me that the next badge to be awarded is Curious badge.  this is my profile: How does vampirism work? A PC in our group was off on his own following a recurring villain of our groups. He was paralyzed where he could move, and the vampire bit his neck. The vampire said "now you get to remain 19 (years old) forever" before he left the cave. This was all done in RP.  We found him later, and healed up his bite marks. Our characters know of vampires, specifically only the one that bit him (recurring villain), but don't really know how vampirism works, because it isn't a super common thing in this world yet.   My question as the player is:  How does RAW vampirism work in 5e?  Is it a disease that develops over time? Is it something where they have to die and be brought back as a vampire? It is a known fact that if $R$ is a UFD, then $R[X_1,X_2,\dots]$ is also a UFD, but there is a subtlety that is making me uncomfortable.  The standard approach essentially goes something along the lines of...let $f$ be a polynomial, so it only involves finitely many variables, say up to $X_1,\dots,X_n$. Then since $R[X_1,\dots,X_n]$ is a UFD, $f$ has a factorization into irreducibles. Then $f$ cannot have a factorization involving indeterminates $X_N$ not in $R[X_1,\dots,X_n]$. If it did, then $f$ would have two factorizations in the UFD $R[X_1,\dots,X_n,\dots, X_N]$, a contradiction.  This seems to assume that the prime factorization in $R[X_1,\dots,X_n]$ is still a prime factorization in $R[X_1,\dots,X_n,\dots,X_N]$. But how can you see that irreducibles in $R[X_1,\dots,X_n]$ are still irreducibles in $R[X_1,\dots,X_m]$ for $m&gt;n$? This does not seem obvious.  Let $R_k=R[X_1,\dots,X_k]$. The explanantion I found is this: Suppose $p\in R[X_1,\dots,X_n]$ is irreducible in $R_n$. Suppose $p=ab$ for $a,b\in R_m$. Evaluating $X_1,\dots,X_n$ at $1$, you get  $$ \bar{p}=\overline{ab}\in R[X_{n+1},\dots,X_m] $$ But $\bar{p}\in R$, which implies that $a$ and $b$ do not involve any variables $X_{n+1},\dots,X_m$. So $a,b\in R_n$, and thus one is a unit, hence a unit in $R_m$.   The part I don't follow is how evaluation at $1$, implies $a$ and $b$ do not have indeterminates other than $X_1,\dots,X_n$. Isn't it possible that some of the higher indexed indeterminates disappear when you evaluate? Suppose for instance $p\in R_1$, and $a\in R_2$ is $a=-X_1X_2+X_2$. Then evaluting at $1$ gives $\bar{a}=0$ so we may not get the desired contradiction since $\overline{ab}=0\in R$? I'm looking for a more politically correct substitute for the expression "poor man's", meaning an inferior improvised or makeshift substitute. Usage examples:   "Guncotton is the poor man's TNT" "Poor man's oyster" (mussels, or coughed up phlegm)    My hesitance is mainly because it is not as gender neutral as perhaps it once was. The closest I can think of is "pauper's", but I don't know many examples of usage. A question about the arctangent addition formula. In the arctangent formula, we have that:  $$\arctan{u}+\arctan{v}=\arctan\left(\frac{u+v}{1-uv}\right)$$  however, only for $uv&lt;1$. My question is: where does this condition come from? The situation is obvious for $uv=1$, but why the inequality?  One of the possibilities I considered was as following: the arctangent addition formula is derived from the formula:  $$\tan\left(\alpha+\beta\right)=\frac{\tan{\alpha}+\tan{\beta}}{1-\tan{\alpha}\tan{\beta}}.$$  Hence, if we put $u=\tan{\alpha}$ and $v=\tan{\beta}$ (which we do in order to obtain the arctangent addition formula from the one above), the condition that $uv&lt;1$ would mean $\tan\alpha\tan\beta&lt;1$, which, in turn, would imply (thought I am NOT sure about this), that $-\pi/2&lt;\alpha+\beta&lt;\pi/2$, i.e. that we have to stay in the same period of tangent.  However, even if the above were true, I still do not see why we have to stay in the same period of tangent for the formula for $\tan(\alpha+\beta)$ to hold. I would be thankful for a thorough explanation. I was going through the fundamental theorem in Number Theory where any non zero integer n can be represented as a product of distinct primes. A related problem with this theorem is to prove that for every such number, there exists a prime $p$ such that $p&lt; \sqrt n$. I was wondering if there is any mathematical proof that no prime $p$ exists for the number $n$ such that $p&gt; \sqrt n$.