Derivative Definition and 1000 Threads

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 am trying to prove an identity for the Lie derivative of a smooth one-form. The identity is: for X, Y smooth vector fields, alpha a smooth one-form, we have: $$L_{[X, Y]}\alpha = [L_X, L_Y]\alpha$$ For anyone familiar with the book, this is exercise 5.26 in the first...

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\...

1. Problem Define a function: for t>=0, f(x,t) = { x for 0 <= x <= sqrt(t), -x + 2sqrt(t) for sqrt(t) <= x <= 2sqrt(t), 0 elsewhere} for t<0 f(x,t) = - f(x,|t|) Show that f is continuous in R^2. Show that f_t (x, 0) = 0 for all x. Then define g(t) = integral[f(x,t)dx] from -1 to 1. Show...

The derivative of secx is $$\d{y}{x} secx =secx tanx $$ But if $$x = \frac{\pi}{3}$$, then $$secx = 2 $$ and the derivative of a constant is 0. And $$sec\frac{\pi}{3} tan\frac{\pi}{3}$$ is equal to $$\frac{3}{2}$$ So what is the derivative of $$secx$$ where $$x = \frac{\pi}{3}$$?

Suppose we have a general timelike congruence of curves with tangent vector field ##V##, then the standard decomposition of the covariant derivative in index form (see e.g. Hawking and Ellis' "Large scale structure of space and time" equation 4.17) is given by $$V_{a;b} = \omega_{ab} +...

Some context for my question: If you have a smooth manifold \mathcal{M} you can define tangent vectors to parametrized paths in the following way: If \mathcal{P}(s) is a parametrized path, then \frac{d}{ds} \mathcal{P}(s) = V where V is the differential operator that acts on scalar fields...

Homework Statement take the derivative of a(t) = b(t)c(t) Homework Equations chain rule The Attempt at a Solution Apply the chain rule: a'(t) = c(t)b'(t) + b(t)c'(t) Is this correct? Thank you.

Homework Statement y = x 2sinx Homework EquationsThe Attempt at a Solution Ok, so If I see an x in an exponent, I would want to use ln to 'bring it out', right? ln y = ln (x 2sinx) = ln x + ln 2sinx = ln x + sinx ln2 now I take the derivative : y'/y = 1/x + cosx ln2 multiply both sides...

Homework Statement Compute Derivative y = xx + sin(x) Homework EquationsThe Attempt at a Solution since I have x in the exponent (x^x), I multiply both sides by ln: ln y = ln xx + ln sin(x) the x in the exponent comes out into the front, right? y'/y = x ln x + ln sin (x) using product...

Hello, I am struggling with these two questions. I think here should be used a quotient rule, but I am not sure how to proceed. a) f(x)=sin$\frac{1}{x}$ b)g(x)=$\frac{1}{sinx}$ Can someone please help. Thanks

Hi. This is not a homework assignment. I am working to get an extrema on a graph that involves a bunch of functions and got stuck on one step: How to get the derivative of: \frac{dy}{dn} = \frac{nc(a+b)}{nc+a} I can't get "n" in a place where I recognize how to get the derivative of it. I...

Homework Statement I am given f(t) = e^-|t| and I found that F(w) = ##\sqrt{\frac{2}{\pi}}\frac{1}{w^2 + 1}## The question says to use the nth derivative property of the Fourier transform to find the Fourier transform of sgn(t)f(t), and gives a hint: "take the derivative of e^-|t|" I also...

[Note from mentor: This thread was originally posted in a non-homework forum, therefore it does not follow the standard homework template.] ------------------------------------------------ Hello. I have some homework to do. I need to make program that finds minimum/maximum of a function od 2...

Would it be a legitimate (valid) proof to use an \epsilon-\delta limit approach to prove the fundamental theorem of calculus? i.e. as the FTC states that if f is a continuous function on [a,b], then we can define a function F: [a,b]\rightarrow\mathbb{R} such that F(x)=\int_{a}^{x}f(t)dt Then F...

Homework Statement In what directions at the point (2, 0) does the function f(x, y) = xy have rate of change -1?D_{u}(f)(a,b) = \bigtriangledown f(a,b)\cdot (u_{1}, u_{2}) f(x,y) = xy (a,b) = (2,0). The Attempt at a Solution \frac{\partial f}{\partial x} = y \frac{\partial f}{\partial y} =...

Homework Statement Find the partial derivative of ## w = xe^\frac {y}{z} ##. Homework Equations N/A The Attempt at a Solution ## \frac{∂f}{∂x} = e^y/z ## ## \frac{∂f}{∂y} = \frac{xe^y/z}{z} ## ## \frac{∂f}{∂z} = (-yz^-2)(xe^yz^-1) ## Are theses correct? Thanks everyone.

In the book "Introduction to Mechanics" by K&K, an increment of a generic time-varying vector is split into two components, ##\Delta \vec{A} _{\perp}## and ##\Delta \vec{A}_{\parallel}##. Their magnitudes are approximated by: $$A \Delta \theta$$ and $$\Delta A$$ respectively. (Where ##\Delta...

Homework Statement D_{u}(f)(a,b) = \triangledown f(a,b)\cdot u D_{(\frac{1}{\sqrt2}, \frac{1}{\sqrt2})}(f)(a,b) = 3 \sqrt{2} where u = (\frac{1}{\sqrt2}, \frac{1}{\sqrt2}) find \bigtriangledown f(a.b) Homework EquationsThe Attempt at a Solution first you change grad f into it's partial...

I have been reading section 3.1 of Wald's GR book in which he introduces the notion of a covariant derivative. As I understand, this is introduced as the (partial) derivative operators \partial_{a} are dependent on the coordinate system one chooses and thus not naturally associated with the...

In chapter 1 of the book "Introduction to Mechanics" by Kleppner and Kolenkow, the derivative of a generic vector ##\vec{A}## is discussed in terms of decomposing an increment in ##\vec{A}##, ##Δ\vec{A}##, into two perpendicular vector vectors; one parallel to ##\vec{A}## and the other...

1. Given a function f(x,y) at (x0,y0). Find the two angles the directional derivative makes with the x-axis, where the directional derivative is 1. The angles lie in (-pi,pi]. 2. f(x,y) = sec(pi/14)*sqrt(x^2 + y^2) p0 = (6,6) 3. I use the relation D_u = grad(f) * u, where u is the...

The bulk modulus B = - V (∂P/∂V). At constant temperature the pressure is given by P= -∂U/∂V, where U is the total energy. We can write B in terms of the energy per particle u = U/N and volume per particle v = V/N : B = v...

Homework Statement Assume that $f(x)$ has two derivatives in $(0,2)$ and $0<a<b<a+b<2$. Prove that if $f(a)\ge f(a+b)$ and $f″(x)\le 0$ $\forall x \in (0, 2)$, then: $$\frac{af(a)+bf(b)}{a+b} \ge f(a+b) \tag 1$$ Homework Equations Below The Attempt at a Solution**MY PROOF:** If $(1)$ is...

Hello, I have this exercise that I can't get the right answer. I have to find derivative of g(x)= (4${x}^{2}$-2x+1)${e}^{x}$ So, what is did is g$^{\prime}$=(8x-2)${e}^{x}$+(4${x}^{2}$-2x+1)${e}^{x}$ My Prof said it is wrong... I am not sure if I have to multiply the brackets or what I did...

Homework Statement I recently searched around SE, and found: http://math.stackexchange.com/questions/1142546/how-to-solve-this-derivative-of-f-proofHomework Equations Below The Attempt at a Solution The answer is interesting. "A function given that $$f(x)=f''(x)+f'(x)g(x)$$ could be an...

I recently searched around SE, and found: [How to solve this derivative of f proof][1] [1]: calculus - How to solve this derivative of f proof? - Mathematics Stack Exchange The answer is interesting. "A function given that $f(x)=f''(x)+f'(x)g(x)$ could be an exponential function, sine...

Homework Statement Find (∂z/∂x) of 6xyz Homework Equations N/a The Attempt at a Solution The correct answer is 6xy(∂z/∂x) but I would like proof of it. I got something different when I tried taking the partial derivative. 6xyz = 6x(yz) = Multiplication rule for derivatives 6(∂x/∂x) +...

Whenever the second order derivative of any physical quantity is related to its second order space derivative a wave of some sort must travel in a medium, why this is so?

i just started with calculus. There was this question my teacher asked us d/dx (x²) = 2x ... eq 1 now we can write 2² = (2+2) 3² = (3+3+3) 4²=(4+4+4+4) . . . n² = (n+n+n+n+...)n times so here d/dx (x²) = d/dx (x+x+x+...)x times so ⇒ d/dx (x) +d/dx(x) +...(x times) = 1+1+1+...(x times) = x ⇒d/dx...

Hi, I'm using partial derivatives to calculate propagation of error. However, a bit rusty on my calculus. I'm trying to figure out the partial derivative with respect to L of the equation: 2pi*sqrt(L/g) (Yep, period of a pendulum). "g" is assumed to have no error. I know I can use the...

Homework Statement Dear Mentors PF Helpers, Here's my question: I see it from my textbook with it solutions copied down below. Wonder is there another way to do it. Thank you for your time.Homework Equations [/B]The Attempt at a Solution

I've attached an image to this post. It essentially shows the equation for the first partial derivative using chain rule, which makes sense. What I'm confused with is how the second partial derivative was formulated. It seems they've simply squared the first partial derivative to find the second...

How do I prove that the parity operator Af(x) = f(-x) commutes with the second derivative operator. I am tempted to write: A∂^2f(x)/∂x^2 = ∂^2f(-x)/∂(-x)^2 = ∂^2f(-x)/∂x^2 = ∂^2Af(x)/∂x^2 But that looks to be abuse of notation..

We know differentiability implies continuity, and in 2 independent variables cases both partial derivatives fx and fy must be continuous functions in order for the primary function f(x,y) to be defined as differentiable. However in the case of 1 independent variable, is it possible for a...

Homework Statement if $$y = \frac{2x^5-3x^3+x^2}{x^3}$$ then $$\frac{dy}{dx} =$$ Homework Equations if $$f(x) = x^n$$ then $$f'(x) = nx^{n-1}$$ The Attempt at a Solution $$\frac{2x^5-3x^3+x^2}{x^3} = \frac{2x^5}{x^3} - \frac{3x^3}{x^3} + \frac{x^2}{x^3}$$ $$ f'(\frac{2x^5-3x^3+x^2}{x^3}) =...

Hello, I try to apprehend the notion of covariant derivative. In order to undertsand better, here is a figure on which we are searching for express the difference \vec{V} = \vec{V}(M') - \vec{V}(M) : In order to evaluate this difference, we do a parallel transport of \vec{V}(M') at point...