Derivative Definition and 1000 Threads

Hi, I am trying to find the minimum root (x) of one formula. For that, I took 2nd derivative and got this equation. \[ 2 \times A \times (\frac{T}{x^3})=0 \] Here A,T,x are greater then 0. I don't know how to proceed further, how to solve it for x? Can you please guide me?

Hello. I bought "Calculus Made Easy" by Thompson and it got me thinking about something I wondered about before. This question is a bit hard for me to articulate, but I'll do my best: When we are trying to find the limit as change in x approaches zero of dy/dx, we take smaller and smaller...

Ref. 'Core Principles of Special and General Relativity' by Luscombe. Apologies in advance for the super-long question, but it's necessary to show my thought process. Let ##\gamma:I\to M## be a smooth curve from an open interval ##I\subset\mathbb{R}## to a manifold ##M##, and let...

I’m am on a path of trying to learn calculus which I should have done long ago. I am making some progress. But I would like to know this... I know what a derivative is. Is differentiation the process of finding a derivative? In other words, when I am finding the derivative can it be said...

Wikipedia defines the derivative of a scalar field, at a point, as the cotangent vector of the field at that point. In particular; The gradient is closely related to the derivative, but it is not itself a derivative: the value of the gradient at a point is a tangent vector – a vector at each...

If $$F(x)=\int_{a}^{b}f(x)dx$$ implies $$F'(x)=\int_{a}^{b}f'(x)dx$$?

Para f (θ) = √3.cos² (θ) + sen (2θ), uma inclinação da reta tangente, uma função em θ = π / 6, é?

Hello there, I have stumbled across further examples to derivatives of multivariable functions that confuse me. Similar to my other thread: https://www.physicsforums.com/threads/partial-derivative-of-composition.985371/#post-6309196 Suppose we have two functions, ## f: R^2 \rightarrow R...

How would someone answer derivative question

It's a detail, but annoying to me: ##{\partial u\over \partial x} = {\partial \phi \over \partial x} \;+ ...## $${\partial u\over \partial x} = {\partial \phi \over \partial x} \;+ ...$$ How do I move up ##\partial u## a little bit so it aligns with ##\partial \phi## ?

This is from Griffiths particle physics, page 360. We have the full Dirac Lagrangian: $$\mathcal L = [i\hbar c \bar \psi \gamma^{\mu} \partial_{\mu} \psi - mc^2 \bar \psi \psi] - [\frac 1 {16\pi} F^{\mu \nu}F_{\mu \nu}] - (q\bar \psi \gamma^{\mu} \psi)A_{\mu}$$ This is invariant under the joint...

I am trying to find the derivative of this problem using the four step process but keep getting stuck when it comes to the third step of f(x+h) - f(x). I do not know what to do once I reach that step. Am I canceling terms out incorrectly? How should I deal with a fraction over a fraction? Any...

Hi, I've been watching lectures from XylyXylyX on YouTube. I believe they are really great ! One doubt about the introduction of Covariant Derivative. At minute 54:00 he explains why covariant derivative is a (1,1) tensor: basically he takes the limit of a fraction in which the numerator is a...

Hello. My understanding of the importance of second derivatives is that they help us to know whether the graph of a function is concave upward or concave downward. In the equation ## f(x) = x^2 + 2x ## we already know from the first derivative, ## f\prime (x) = 2x + 2 ##, that the graph is...

a) I found this part to be quite straight forward. From the Parallel transport equation we obtain the differential equations for the different components of ##X^\mu##: $$ \begin{align*} \frac{\partial X^{\theta}}{\partial \varphi} &=X^{\varphi} \sin \theta_{0} \cos \theta_{0}, \\ \frac{\partial...

I was solving a problem for my quantum mechanics homework, and was therefore browsing in the internet for further information. Then I stumbled upon this here: R is the rotation operator, δφ an infinitesimal angle and Ψ is the wave function. I know that it is able to rotate a curve, vector...

Hello everyone, I'm stuck doing this problem, I've tackled the partial derivative but i can't figure out the derive for x component part, i solved the partial derivative part, i came to this result: What do can i do from here on, thank you!

I am unsure how to go about this. I tried following the suggestion blindly and end up with with some cumbersome terms that are not the answer. From what I understand the derivative at each point would equal to T? Answer: I just can seem to get to this. I think I'm there but can't get it in...

I read in one book proving one nature of variation(variation of high-order derivative). It writes that "##\delta(F^{(n)}) = F^{(n)} - F_0^{(n)} = (F - F_0)^{(n)} = (\delta F)^{(n)}##". But I don't understand where this ##F_0## comes out from.

I'm thinking about that for 1 hour. But, I could not do it.

Hello! This if from a physics paper but I will write it as abstract as I can. We have a function ##f(g(a),a)## and we know that f is minimized with respect to g for any given a i.e. $$\frac{df}{dg}|_a=0$$ As this is true for any a, we have $$\frac{d}{da}\frac{df}{dg}|_a=0$$ from which we get...

Did I calculate this properly?

Hello everybody, could you help me with this problem please? I have to find a derivative in x0 of this function (without using L'Hospital's rule): I used the definition , but I don't know what to do next. Thank you.

I have no idea how this derivative was taken.

Below are plots of the function ##e^{0.25(x-3)^{-2}} - 0.87 e^{(x-3.5)^{-2}}## The first plot is for real values. It has a minimum at the red dot. The second plot has in its argument the same real part as the red dot, but has the imaginary part changing from -0.3 to 0.3. It shows the resulting...

Hello- In the attached screenshot from my textbook, I am trying to understand how they get from equation 6.5 to 6.5a. I have attached my attempt to solve it, but I am stuck evaluating the left side. I do not see how to get their result. Relevant information: k, T_w, T_inf, h and L are all...

My try: ##\begin{align} \dfrac{d {r^2}}{d r} \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \tag1\\ \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \dfrac{1}{\dfrac{d r^2}{d r}}=\dfrac{p-a\cos\theta}{r} \tag2\\ \end{align}## By chain rule...

While working at home during the COVID-19 pandemic I've taken to seeing if I can still do math from undergrad (something I do once in a while to at least pretend my life isn't dominated by excel). So to that I've been reviewing partial derivatives (which I haven't really thought about in a good...

When I type in this: D [ Re[ Exp[u + 10*I] ], u ] /. u->0.5 I get this output: Of course, I could just put the Re outside and the D inside, but it would be nice to know what is wrong with the above. What's with the Re' in the output?

Hi guys, suppose we have a function ##C(x, y)## into the real numbers. Suppose also that ##y=y(x)##, i.e. ##y## is a function of ##x##. Now in my script, I have a term ##\nabla_x C(x_0, y(x_0)) ##. From my point of view, this means that you take the partial derivative of ##C(x,y)## with...

I apologize: despite my verbosity, this is, I hope, a simple question.) Consider the following relationship between a rotating reference frame and an inertial reference frame (both Bold), through a rotation matrix: (the superscript is to designate the rotating frame e(1) and the I is for the...

If $$f(x)=\frac{3x^2-5}{x+6}$$ then f(0) + f'(0) is ... A. 2 B. 1 C. 0 D. -1 E. -2 What I did: If $$f(x)=\frac{u}{v}$$ then: u =$$3x^2-5$$ → u' = 6x v = x + 6 → v' = 1 f'(x) =$$\frac{u'v-uv'}{v^2}$$=$$\frac{6x(x+6)-(3x^2-5)(1)}{(x+6)^2}$$ f(0) + f'(0) = $$\frac{3(0^2)-5}{0+6}$$ +...

Hello, I would need some help in calculating the derivative of the function T_el in the attached image. I want to calculate d T_el /d yd, where yd is the variable and it appears in the term I called A_elSide. Its expression is again in the image. Numbers you see are not important.Just to...

I don't understand how to use output from an NPT molecular dynamics simulation to compute a thermodynamic derivative. I need to compute this (where "d" is a partial derivative, "T" is a subscript that means, "at constant temperature," and "E" is internal energy): -(dE/dV)T I have a simulation...

My attempt I calculated the partial derivatives of n wrt P and T. They are given below. ##\frac {\partial n}{\partial P} = \frac{nb -1}{\left(2an-Pb-3abn^2-kT\right )}## ##\frac {\partial n}{\partial T}= \frac {nk}{\left(2an-Pb-3abn^2-kT \right ) }## I know that if the partial derivative is...

Hi all. A problem has arisen whereby I need to maximize a function which looks like $$ f(A) = \mathbf{w}^T \left[\int_0^t e^{\tau A} M e^{\tau A^T} d\tau \right]^{-1} \mathbf{w} $$ with respect to the nxn matrix A (here, M is a covariance matrix, so nxn symmetrix and positive-definite, w is an...

Let us suppose we have a functional of f such that ##f=f((\vec{r}(t),t)## where ##\vec{r}(t) = a(t)\vec{x}(t)##. I am trying to derive an equation such that $$\left.\frac{\partial}{\partial t}\right|_r = \left.\frac{\partial }{\partial t}\right|_x + \left.\frac{\partial \vec{x}}{\partial...

Can you take any non invariant quantity like components and take the covariant derivative of them and arrive at an invariant tensor quantity? Or are there limits on what you can make a tensor?

So just based on the cauchy riemann theorem, I think: Ux = 2 = Vy = 2xy, so f(z) is differentiable on xy = 1, and also that Vx = y^2 = -Uy = 0. That doesn't make sense to me because if 0 = y^2, then y = 0, yet that wouldn't satisfy xy = 1, would it? Furthermore, I'm not sure how I would...

Hello, In second-order derivative test, the test is inconclusive when ##f''(c)=0##, so we had to generalize to higher-order derivative test. I was wondering how such tests can be generalized and derived? For example, how can I prove that ##f(x)=x^4## have minimum at 0? Bagas

I read recently that Einstein initially tried the Ricci tensor alone as the left hand side his field equation but the covariant derivative wasn't zero as the energy tensor was. What is the covariant derivative of the Ricci tensor if not zero?

This is from QFT for Gifted Amateur, chapter 14. We have a Lagrangian density: $$\mathcal{L} = (D^{\mu}\psi)^*(D_{\mu}\psi)$$ Where $$D_{\mu} = \partial_{\mu} + iq A_{\mu}(x)$$ is the covariant derivative. And a global gauge transformation$$\psi(x) \rightarrow \psi(x)e^{i\alpha(x)}$$ We are...

I tried to calculate the directional derivative but the answer that I found was 194.4 but the answer marked in the book was 540. I tried a lot but couldn't understand what my mistake was. Please let me know what mistake I did.

I will start from the meaning of increasing function. A function is said to be increasing function if for x < y then f(x) ≤ f(y). Is this correct? Then f(x) is increasing function if f'(x) ≥ 0. Is this correct? Lately I encounter the term "monotonic increasing". What is the difference between...

Greetings, I am struggling with an exercise to the Langevin equation. Suppose we are given the following differential equation for a particle's 1D time-dependent momentum ##p(t)##: $$\text{d}p = -\gamma p \text{d}t + F(r)\text{d}t + \sqrt{C\gamma}\text{d}W $$ with a constant ##C##, a...

Moved from technical math section, so missing the homework template Summary:: Find a general formula for the nth derivative Hi everyone! How would I approach and answer a Q such as this I began by rewriting the expression in a different form, then used chain rule to each given term I...

I tried to derive the right hand side of the Radon-Nikodym derivative above but I got different result, here is my attempt: \begin{equation} \label{eq1} \begin{split} \frac{\mathrm d\mu_{\Theta\mid X}}{\mathrm d\mu_\Theta}(\theta \mid x) &= f_{\Theta\mid X}(\theta\mid x) \mathrm \space...

i compute the partial derivative, the vector that i have to use the one in the text or w=(2/(5^(1/2)), 1/(5^(1/2))) using the last one i get minus square root of five , if i don't divide by the norm the answer should be B. i don't understand what D means

The function should use (r,z,t) variables The domain is (0,H) Since U is not dependent on angle, then theta can be ignored in the expression for Laplacian in cylindrical coordinates(?)

Here is this week's POTW: ----- Suppose $f : (a,b) \to \Bbb R$ is a convex function. Show that $f$ is differentiable at all but countably many points and the derivative is nondecreasing. ----- Remember to read the...