Derivative Definition and 1000 Threads

I'm looking for a book to self-study this summer before my last course of Bachillerato (A Levels or High School in other countries). My performance in Mathematics has only improved and I've just been given the maximum mark this last term (10/10 or A+). This has motivated me a lot to keep...

Homework Statement If f(x+y) = f(x)f(y) for all x and y and f(5) = 2, f'(0)=3, then f'(5) is a) 5 b) 6 C) 0 d) None of these Homework Equations Don't know which equation to apply. I was thinking of Rolle's Theorem and mean value theorem here but it is not helping. The Attempt at a Solution...

Let G be a non-empty open connected set in Rn, f be a differentiable function from G into R, and A be a linear transformation from Rn to R. If f '(a)=A for all a in G, find f and prove your answer. I thought of f as being the same as the linear transformation, i.e. f(x)=A(x). Is this true?

The question suggests that r(t) = (x(t),y(t),z(t)) is a position vector along some curve where t goes from negative to positive infinity. Now suppose t has been chosen so that 1 = the dot product of dr/dt and dr/dt. Show that 0 = the dot product of dr/dt and d^2r/dt^2. I have attempted to...

Hey guys and gals, I'm taking an online physics course just to kind of learn the basics before I take the real thing this summer. The course is from OpenYale for those interested (oyc.yale.edu). Anyways, the professor was talking about some formulas for finding ##\vec{r}(t) = r(i(cos\omega t) +...

In my textbook (see attached picture) there appears a functional derivative, but I honestly don't know how to evaluate a quantity like this. What should I do? I have tried to google but all I could find was how to take functional derivatives, where polynomials appeared under the integral, while...

Homework Statement For the equation ∇ x E = -∂B/∂t I took the curl of both sides to get ∇ x (∇ x E) = ∇ x -∂B/∂t I feel like it'd be very wrong to pull out the time derivative. Am I correct?

I have attached an image of a function that I fit to a scatter plot, and I would like to know if there is a term for the point on the function at which the slope transitions from being less than -1 to greater than -1. I have highlighted this point approximately in yellow...

Homework Statement The website says this: "It is Linear when the variable (and its derivatives) has no exponent or other function put on it. So no y2, y3, √y, sin(y), ln(y) etc, just plain y (or whatever the variable is). More formally a Linear Differential Equation is in the form: dy/dx +...

Homework Statement Given two curves y=f(x) passing through (0,1) and ##g(x)=\int\limits_{-\infty}^xf(t)dt## passing through (0,1/n). The tangents drawn to both curves at the points with equal abscissae intersect on the x-axis. Find y=f(x). Homework Equations None The Attempt at a Solution...

I'm hoping someone can clarify for me, I have seen the following used: \frac{\partial}{\partial g^{ab}}\left( g^{cd} \right) = \frac{1}{2} \left( \delta_a^c \delta_b^d + \delta_b^c \delta_a^d\right) I understand the two half terms are used to account for the symmetry of the metric tensor...

Homework Statement [/B] use richardson extrapolation to estimate the first derivative y=ln(x), x=5 using steps of 2, 1, 0.5. Four decimal points. obtain true relative error for the last estimate and comment on its value. Homework Equations [/B] deriv ln(x)=1/xThe Attempt at a Solution I know...

Homework Statement I am trying to prove without using the mean value theorem that two different functions with the same derivative differ only by a constant. Is it possible to do this without the mean value theorem? If so, would someone help guide me towards the right solution. Homework...

For a direction determined by $dx=2dy=-2dz$, find the directional derivative of $F=x^2+y^2+z^2$ at P(1,2,1) I had no problem getting the gradient of F and evaluating it at P but when I take the directional derivative I'm stuck! I don't know how come up with a unit vector that should be dotted...

I can't convince myself whether the following functional derivative is trivial or not: ##\frac \delta {\delta \psi(x)} \big[ \partial_x \psi(x)\big],## where ##\partial_x## is a standard derivative with respect to ##x##. One could argue that ## \partial_x \psi(x) = \int dx' [\partial_{x'}...

Homework Statement Solve ∂v/∂θ and ∂v/∂r. (refer to attached image for equations) Homework Equations Refer to attached image. note that the velocity is expressed in cylindrical coordinates and attention must be paid to the directional unit vectors eθ and eρ.[/B] The Attempt at a Solution...

I was reading through hobson and my notes where the covariant acts on contravariant and covariant tensors as \nabla_\alpha V^\mu = \partial_\alpha V^\mu + \Gamma^\mu_{\alpha \gamma} V^\gamma \nabla_\alpha V_\mu = \partial_\alpha V_\mu - \Gamma^\gamma_{\alpha \mu} V_\gamma Why is there a minus...

Hi all, I have a question concerning the derivative of the formula of the number of nuclei. I hope I've posted this in the right section, I'm new here :P. Anyway, in the question, the given values are: At a certain time t, there is an amount of radioactive Br-82. The activity A is 7.4*1014 Bq...

Acting upon a vector say, so it is defined as: ##\frac{d}{d\lambda}V^{u}+\Gamma^{u}_{op}\frac{dx^{o}}{d\lambda}V^{p}=\frac{DV^{u}}{D\lambda}## And this can also be written in terms of the covariant derivative, ##\bigtriangledown_{k}## by ##\frac{DV^{u}}{D\lambda}=\frac{d x^{k}}{d \lambda}...

What does it mean to say that ##\displaystyle\frac{d }{d x}\int f(x)dx = f(x)##? Does this somehow relate to the fundamental theorem of calculus? If so, how?

what is the derivative of x! ??

SOLVED 1. Homework Statement Find polynomial of least degree satisfying: p(1)=-1, p'(1)=2, p''(1)=0, p(2)=1, p'(2)=-2 Homework Equations In general, a Hermite Polynomial is defined by the following: ∑[f(xi)*hi(x)+f'(xi)*h2i(x)] where: hi(xj)=1 if i=j and 0 otherwise. Similarly with h'2...

Homework Statement I was working on a physics problem that involves integrals and I stumbled upon this: m*g*sin θ − k*x = m*dv/dt =m*v*dv/dx → (m*g*sin θ − k*x)/m =v*dv/dx →∫[(m*g*sin θ − k*x)/m]dx =∫v*dv Notice that we multiplied by dx on both sides and dx has been canceled from the right...

Homework Statement [/B] find directional derivative at point (0,0) in direction u = (1, -1) for f(x,y) = x(1+y)^-1The Attempt at a Solution grad f(x,y) = ( (1+y)^-1, -x(1+y)^-2 ) grad f(0,0) = (1, 0) grad f(x,y) . u = (1,0).(1,-1) = 1. seems easy but markscheme says I am wromg. It says...

Homework Statement Derivative of P(x) = 0.2 -0.125e^(0.005x) for X=1 P(x) is the population in Millions and X is the year where 0 = 2000 I have to explain the answer I am getting at the end. Homework EquationsThe Attempt at a Solution d/dx (0.2 -0.125e^(0.005x)) I find -0.000625*(1,00501)^x...

Suppose someone gives you a rectangular sheet of length a and width b (so b ≤ a). You make a topless box by cutting out a square with length x out of each corner and folding up the sides. How should you cut the sheet so as to maximize the volume of your box?

Homework Statement [/B] I have to derivate f(x) f(x) = 1/x +1/(x+1) Answer is = -1/x2 - 1/(x+1)2 I can't seem to get to that answer :( thank you Homework EquationsThe Attempt at a Solution f(x) = (1/x +1/(x+1))' = (2x+1)/x(x+1)' = ((2x+1)' * (x(x+1)-(2x+1)*(x(x+1)')/x2(x+1)2...

Homework Statement f(x) = 1/ln (10-x) -- I would assume it to be a fairly simple equation, but I am screwing it upHomework Equations What is f'(x)?The Attempt at a Solution f'(x) = (ln (10-x))^-1 = -(ln (10-x))^-2 * -1 * 1/(10-x) -- 2 negatives cancel out = 1/(10-x) (ln(10-x))2 --...

Homework Statement So I'd rather not type out the whole equation I am differentiating with respect to t... Sorry admins. My written work is on the image. I just want to make sure my work is correct. Homework Equations The equation is a differential characteristic equation with cos and sin...

I'm trying to prove that ##\sqrt{-g}\bigtriangledown_{\mu}v^{\mu}=\partial_{\mu}(\sqrt{-g}v^{\mu}) ## So i have ##\sqrt{-g}\bigtriangledown_{\mu}v^{\mu}=\sqrt{-g}(\partial_{\mu}v^{mu}+\Gamma^{\mu}_{\mu \alpha}v^{\alpha}) ## by just expanding out the definition of the covariant derivative...

Homework Statement substitute the given power series below into ODE y'' -4y=0 to verify it is a solution Homework Equations y=∑ 2n xn / n! n=0 y''-4y=0 The Attempt at a Solution I have absolutely no idea how start.

What's the difference between derivative of displacement with respect to time and derivative of position with respect to time ?I have read derivative of displacement with respect to time is velocity and derivative of position with respect to time is also velocity,so what's the difference? I...

<<Moderator note: Remember that filling in the complete homework template is mandatory in the homework forums. This thread has not been deleted due to containing relevant replies.>> 1. Homework Statement ((x)^(1/x))' Homework Equations This probably isn't overly dificult, but it has got me...

Homework Statement Hello, I had a few derivative of the natural logarithm functions questions. It seems like it should be fairly straightforward, but I am turning it into a pig’s ear.On my honor, none of these are problems on an assessment per se, however, they are not materially different than...

I'm looking at the deriviation of Einstein's equation via applying the principle of least action to the Hilbert-Einstein action. I'm trying to understand the vanishing of a term because it is a total derivative: http://www.tapir.caltech.edu/~chirata/ph236/2011-12/lec33.pdf, equation 19. My...

Hi, I'm reading Ogata's Modern Control Engineering, and when he talks about the representation of a differential equation in state space he divides the method in two. The first one is when the input of the differential equation involves no derivative term, for example: x'(t)+x(t)=u(t) The...

Hi, I have some points say, 100 points which come from a periodic tube profile, i.e., (z,r), where z and r are the axial and radial coordinates, respectively. Now, I need to calculate the first derivative at each point. Could you please help me in this regard? Cheers

I am getting a little confused on which error propagation to use: I am looking to calculate the error in B*Cos(θ) , for the vertical axis of a williamson hall plot. where B is fwhm of a peak with it's own error and cos of the bragg angle I am unsure of whether i need to use partial derivative...

Homework Statement I will post a picture of the problem and then the second picture will be my work. The problems are the first two. Homework EquationsThe Attempt at a Solution I didn't know how to do this at first so I don't know if I am doing it correctly now. Also I don't know the correct...

Homework Statement I posted a picture of it and my attempt it is number 3 Homework EquationsThe Attempt at a Solution I tried using log properties and I am not sure what went wrong and how to arrive at the correct answer. Mod note: Messy, disorganized image deleted.

Homework Statement Hello all, thank you for the help in advance. It's a two-sided derivative problem, for lack of a better term, and I appreciate all hints or help. If we have a function y so that y=bx for all x<0, and y= x^2-13x for all x> or = 0, for what value of b is y differentiable at...

Homework Statement So the test is to take the determinant (D) of the Hessian matrix of your multivar function. Then if D>0 & fxx>0 it's a min point, if D>0 & fxx<0 it's a max point. For D<0 it's a saddle point, and D=0 gives no information. My question is, what happens if fxx=0? Is that...

Dear All, Please see the image below in attachment where Energy is function of K. I want to understand how is it possible to understand the last expression ( dE = ? ). Additionally, what is the difference between curly and derivative (d) sign ? Many thanks to the mentors on this forum Best wishes

Okay guys, this is driving me absolutely nuts. I'm working on finding derivatives using the product and quotient rules and the book will sometimes simplify the problem before finding the derivative but sometimes wont and I don't understand why. For example: The function y = (v3-2v√v)/v The book...

Is the time derivative of a curl commutative? I think I may have answered this question... Only the partial time derivative of a curl is commutative? The total time derivative is not, since for example in cartesian coordinates, x,y,and z can themselves be functions of time. In spherical and...

Homework Statement Given that the surface x^7y^2+y^4z^6+z^8x^8+9xyz=12 has the equation z=f(xy) in a neighbourhod of the point (1,1,1) with f(x,y) differentiable, find the derivatives. df/dx (1,1) = ? d^2f/dx^2 (1,1) = ? Homework EquationsThe Attempt at a Solution df/dx (1,1) I got -24/23 or...

Homework Statement a) Show that the function f(x,y)=\sqrt[3]{xy} is continuous and the partial derivatives f_x and f_y exist at the origin but the directional derivatives in all other directions do not exist b) Graph f near the origin and comment on how the graph confirms part (a). 2. The...

Hi, I've learned that material derivative is equal to local derivative + convective derivative, but can't seem to find out which way the convective derivative acts, like for example in velocity fields: The equation my teacher gave us was (with a and v all/both vectors): Acceleration = material...

Homework Statement So the first part asks to prove the time derivative of kinetic energy is dT/dt=F dot product v which I did not problem. but then the second part of the problem asks to prove that if the mass is changing with time then the time derivative of d(mT)/dt=F dot product m and I'm...

When can I do the following where ##h_{i}## is a function of ##(x_{1},...,x_{n})##? \frac{\partial}{\partial x_{k}}\frac{\partial f(h_{1},...,h_{n})}{\partial h_{m}}\overset{?}{=}\frac{\partial}{\partial h_{m}}\frac{\partial f(h_{1},...,h_{n})}{\partial x_{m}}\overset{\underbrace{chain\...