2
votes
1answer
56 views

Understanding the notation of a book when derivating

I'm trying to understand the notation that the book uses. The book says $(1)$ $y=a\cdot \sin x$ and then the derivate of $(1)$ is $(2)$ $\frac{d^2y}{dx^2}=-a \cdot \sin x$ I don't get what to do ...
1
vote
0answers
31 views

Weird derivative computation

I found the following formulas in a control theory textbook : $$s(x,t)=\left(\frac{d}{dt}+\lambda\right)^{(n-1)}\varepsilon $$ where $\varepsilon(t)=T\left(\frac{e(t)}{p(t)}\right)$ and ...
0
votes
2answers
45 views

What is this lower number?

I was taught that the lower number in math would be the base, but you can't have base 0 (can you?) I'm looking at some derivatives and it looks something like this. $$x^2_0$$ Sorry for the stupid ...
2
votes
0answers
37 views

Understanding notation with regards to tangent derivatives.

I am currently reading a paper on monge-ampere equations, and in one part the author does as follows. Let $\Omega,\Omega^*$ be two uniformly convex subsets of $\Bbb R^n$, and let $h\in C^{2,1}(\Bbb ...
1
vote
2answers
50 views

What does d f(t,x) = 0 mean?

A differential equation that can be written in the form $d\phi(t, x) = 0$ for some continuous and differentiable function $\phi(t, x)$ is called exact. What does $d\phi(t, x) = 0$ mean?
3
votes
1answer
85 views

Is this notation good for the chain rule derivative?

When we take this derivative, for example: $$y = \log(\sin x)$$ We call $u = \sin x$, so we have: $$\frac{dy}{dx} = \frac{d y}{du}\frac{du}{dx} = \frac{1}{u}\cos x = \frac{\cos x}{\sin x}$$ But for ...
2
votes
2answers
92 views

What does $\frac{d^2}{dx^2}$ stands for

I would like to know what is $\frac{d^n}{dx^n}$. I think it stands for dervation of $n-$th order but I am not sure.
2
votes
5answers
59 views

The Notation for Derivatives

"The derivative of a sum is the sum of derivatives" Above theorem can be mathematically expressed as: $$h'(x)=f'(x)+g'(x)$$ where $f(x)$ and $g(x)$ are two differentiable functions. What is the ...
4
votes
1answer
65 views

Uncomfortable using Leibniz notation for the chain rule.

I am working through the following solved problem which uses separation of variables to get two ODEs. The problem is to show that $$\frac{1}{\sin\theta ...
0
votes
2answers
53 views

Why does $d$ mean?

What do the $d$'s mean? I've seen them in other formulas as well.
3
votes
5answers
118 views

Chain Rule: Is the notation $\frac{dy}{du} \cdot \frac{du}{dx} = \frac{dy}{dx}$ accurate?

My question is if it is okay / mathematically rigorous to write the Chain Rule like that (the Leibniz way). I thought that $dx$, etc. do not follow the rules of algebra and cannot be treated as such. ...
2
votes
3answers
95 views

Why are there so many notations for differentiation?

There are so many notations for differentiation. Some of them are: $$ f^\prime(x) \qquad \frac{d}{dx}(f(x))\qquad \frac{dy}{dx}\qquad \frac{df}{dx}\qquad D f(x)\qquad y^\prime\qquad D_x f(x) $$ Why ...
1
vote
3answers
30 views

Notation for function compositions/derivatives

When given $(f \circ g)'(0)$, does it mean to compose the 2 functions first, then take the derivative of the composed functions and evaluate it at $0$, or take the derivative of $g$ first and evaluate ...
0
votes
1answer
64 views

What does the notation $Dx(a-bi-x)$ mean?

Sorry for completely newbie question, but could not the answer anywhere else. On the Wikipedia page about "Steiner inellipse" there is the following notation equating roots of cubic polynomial and ...
1
vote
1answer
38 views

On notation for derivative of an n-Dimensional Gaussian

How to we represent the derivative of a n-D gaussian function defined by $g(\mathbf{x}) = \dfrac{1}{\sqrt{2 \pi \left|\Sigma\right|} } \exp^{-\dfrac{1}{2}({\mathbf{x}-\boldsymbol\mu}) ^\top \Sigma ...
4
votes
3answers
686 views

The difference between $\Delta x$, $\delta x$ and $dx$

$\Delta x$, $\delta x$ and $dx$ are used when talking about slopes and derivatives. But I don't know what the exact difference is between them.
1
vote
1answer
26 views

Multiplication formula for Lie derivative

Let $U\in\mathbb{R}^n$ be an open set, and let $f_1,f_2\in C^1(U)$. Prove that $$L_v(f_1f_2)=f_1L_vf_2+f_2L_vf_1$$ Suppose $f_1,f_2:U\rightarrow\mathbb{R}$. Let $p\in U$. We have ...
1
vote
2answers
83 views

How would one prove $[f,[\nabla^2,f]]=-2(\nabla f)^2$?

How would one prove this equation: $$[f,[\nabla^2,f]]=-2(\nabla f)^2 $$ And I'm confused that $\nabla f\nabla f$ equals $(\nabla f)^2$ or $\nabla(f\nabla f)$.
1
vote
2answers
132 views

Notation regarding different derivatives

I am currently reading up on partial derivatives and differentials in general. And there are a few points that seem unlcear to me (notation-wise). For example, if $f:\mathbb R\to\mathbb R,x\mapsto ...
2
votes
3answers
92 views

Using both Leibniz' notation and prime-notation for a derivative

I am presented with the following task: "Assume that the function $f(x)$ has the derivative $f'(x) = \frac{1}{x}$ and that $f$ is one-to-one. If $y = f^{-1}(x)$, show that $\frac{dy}{dx} = 1$. The ...
1
vote
0answers
56 views

Abbreviation for $n$ times differentiable, with $n$th derivative bounded?

Are there convenient abbreviations in use for the following sets? The set of functions which are $n$ times differentiable, with first $n-1$ derivatives continuous (obviously the last part is ...
2
votes
2answers
163 views

Does differentiation symbol need parentheses or?

Suppose I have this expression: $$\frac{d}{dx}(e^{x})^2 + 6$$ Does it mean to differentiate $6$ too or just the first term? This is an exercise on a calculus course that I'm doing on Coursera. ...
2
votes
2answers
107 views

What exactly does the $d$ represent in $\frac{d}{dx}$?

When taking the derivative, such as $\frac{d}{dx}$, what exactly does the $d$ represent? The best answer so far is in for example $\frac{dy}{dx}$, the $d$ stands for change in and what follows the ...
8
votes
3answers
239 views

Is $d^2y/dx$ a valid mathematical notation?

I have often seen "the second derivative of y with respect to x" written as $${d^2y\over dx^2},$$ but I don't understand the reason for this notation. I have always seen it written as $${d^2y\over ...
4
votes
4answers
231 views

Leibniz notation for high-order derivatives

What is the reason for the positioning of the superscript $n$ in an $n$-order derivative $\frac{d^ny}{dx^n}$? Is it just a convention or does it have some mathematical meaning?
4
votes
1answer
118 views

Notation of derivatives…

I asked my teacher the difference between this notations. (1) $$\frac{dy}{dx}$$ (2) $$\frac{\delta y}{\delta x}$$ (3) $$\frac{\Delta y}{\Delta x}$$ He told me that there is no difference. I really ...
2
votes
1answer
52 views

Notation for function being differentiable at a certain point

This question describes a notation for a function $f(x)$ being (continuously) differentiable on some domain $A$. Often, I see the requirement that some function $f(x)$ be differentiable only (or ...
5
votes
1answer
111 views

Why do we write second derivatives like $\frac{d^2x}{dt^2}$ [duplicate]

Why do we write the second derivative of $x$ with respect to $t$ as $\frac{d^2x}{dt^2}$? It's never been explained to me, and I've never found a particularly good explanation. What's up with this ...
4
votes
5answers
119 views

The derivative of $e^x$

We all know that the derivative of $e^x$ equals $e^x$. I found this notation on Wikipedia: $$\dfrac{d}{dx} e^x = e^x$$ Why isn't this expression $$\dfrac{dy}{dx} e^x = e^x$$ Is it because ...
5
votes
4answers
154 views

What is the use of, and intuition behind, writing $\frac{d^2}{dx^2}$ for the second derivative?

Is it possible to take a second derivative without taking the first derivative before? Why do we multiply the $d$ and $dx$ operators? Like, does $\dfrac{d^2}{dx^2}$ really mean $\dfrac{d}{dx} \cdot ...
2
votes
2answers
47 views

How can partial derivatives feature in the definition of a function?

I have a map $f(t,g,h)$ where $f:[0,1]\times C^1 \times C^1 \to \mathbb{R}.$ I want to define $$F(t,g,h) = \frac{d}{dt}f(t,g,h)$$ where $g$ and $h$ have no $t$-dependence in them. So $g(x) = t^2x$ ...
-2
votes
4answers
254 views

Derivative of product notation?

Presume $f(x,y)$ is a continuous function. How would I take the derivative of $$\prod_{x=1}^N f(x,y)$$? Edit: derivative with respect to $x$, that is.
6
votes
4answers
326 views

If $f(x)\to f(a)$ when $x\to a$, why don't we denote it as $\displaystyle \lim_{x\to a}f(x)\to f(a)$?

If $f(x)\to f(a)$ when $x\to a$, why don't we denote it as $$ \lim_{x\to a}f(x)\to f(a) $$ instead of $$ \lim_{x\to a}f(x) = f(a)? $$ I need a comprehensible explanation for a newbie like me!
0
votes
2answers
148 views

Leibniz notation - how to get $dx$ out of a derivative $v = \frac{dx}{dt}$

I know that velocity equals $v = \frac{dx}{dt}$ which is writen in Leibniz notation. How can i get $dx$ out of it in a proper way? I don't like it when people say that i should just multiply ...
0
votes
3answers
2k views

Differentiation, using d or delta

Are the symbols $d$ and $\delta$ equivalent in expressions like $dy/dx$? Or do they mean something different? Thanks
1
vote
1answer
42 views

Notation issue regarding differential equations

I am given the following problem: Find a basis of solutions for the equation: $u^{iv} + 2u'' + 3u = 0$ The notation is an exact duplicaticate of what our professor used in his notes. Does anybody ...
0
votes
2answers
49 views

What is $y^{(n)}$, and how do I find it?

The question : Find $y^{(n)}(x)$ if $y(x)=\frac{1}{2-x}$... there is no explanation for $y^{(n)}$ in my textbook...can you explain this to me? First I tried to find the derivative of $\frac{1}{2-x}$ ...
0
votes
1answer
63 views

Partial derivative notation: $\left.\frac{\partial \cdot}{\partial\cdot} \right|_{u=T}$

Let $\displaystyle \ \ B(t,T):=\int_t^T f(t,s)ds$, where $f(.,.)$ is a stochastic process whose solution we don't know. My lecture slides make the claim that: $$f(t,T) = \frac{\partial ...
0
votes
2answers
137 views

Notation of derivative

When I studied derivative, I sometimes saw notation $\frac{d}{dx}2x=2$ and sometimes $\frac{\partial}{\partial x}2x=2$. What is the definition and difference between those notations?
1
vote
3answers
124 views

Help with odd partial derivatives in velocity $\bar v^2 = \dot x ^2+\dot y^2$

I am doing a physics -course Tfy-0.2061. My teacher claims that this is velocity squared, $\bar v^2 = \dot x ^2+\dot y^2$. I cannot understand why it is not $\bar v^2 = (\dot x +\dot y)^2$. If ...
1
vote
2answers
441 views

What is the difference between $d$ and $\partial$?

After seeing the following equation in a lecture about tensor analysis, I became confused. $$ \frac{d\phi}{ds}=\frac{\partial \phi}{\partial x^m}\frac{dx^m}{ds} $$ What exactly is the difference ...
2
votes
2answers
144 views

derivative at a given point

I often see the following: $$ \left. \frac{\partial q}{\partial \alpha} \right|_{\alpha = 0} $$ Where $q$ is a function $q(q', t, \alpha)$. Is that just the same as that? $$ ...
6
votes
1answer
287 views

notation of derivation in differential geometry

I can't wrap my head around notation in differential geometry especially the abundant versions of derivation. Peter Petersen: Riemannian Geometry defines a lot of notation to be equal but I don't ...
2
votes
3answers
61 views

Clarification regarding a derivative symbol

I came across the following expression: $$\frac{\partial^{i_1+\cdots+i_m}P(x_1,\ldots,x_m)}{\partial x_1^{i_1}\cdot\cdot\cdot \partial x_m^{i_m}}$$ for $P(x_1,\ldots,x_m)$ a polynomial in $m$ ...
2
votes
4answers
1k views

Second order partial derivatives - notation

I have seen both of these used, and people around me seem to disagree, so which one is correct: (first derivative with respect to x, then y): (1) $$\frac{\partial }{\partial y}(\frac{\partial ...
3
votes
2answers
134 views

What is the difference between these two derivative expressions?

is there a difference between $\frac{\partial^2 }{\partial x^2}$ and $(\frac{\partial }{\partial x})^{2}$? I have to tell if a differential equation is linear, and $(\frac{\partial }{\partial x})^{2}$ ...
4
votes
2answers
1k views

How do you pronounce (partial) derivatives?

I am not an English speaker that is why I asked this question. In addition, I think english.stackexchange.com is not the proper place to ask this because (I am so sorry) I don't think most of them ...
0
votes
2answers
396 views

Is this the notation you use?

I've noticed that my terminology is a bit haggard. I do math on my own so I'm not entirely sure how everyone else refers to things and so I need a check. so is this correct: $\lim\limits_{\delta x ...