Differentiation Definition and 1000 Threads

Let $E\subset\mathbb{R}^n$ and $f: E\rightarrow\mathbb{R}$ be a continuous function. Prove that if $a$ is a local maximum point for $f$, then either $f$ is differentiable at $x = a$ and $Df(a) = 0$ or $f$ is not differentiable at $x = a$. Deduce that if $f$ is differentiable on $E^o$, then a...

Homework Statement I'm trying to take the derivative of the following integral \frac{d}{d V} \int_0^t{V(\tau)}d\tau Homework Equations FTC will probably be a part of it. The Attempt at a Solution I always get confused when I'm taking the derivative of an integral. I know the answer is...

Hi guys! Let \left \{ B_{t} \right \}_{t\in \mathbb{R}} be a one - parameter family of compact subsets of \mathbb{R}^{3} with smooth (manifold) boundary (e.g. one - parameter family of closed balls). In my context, each B_{t} belongs to a different constant time slice of Minkowski space - time...

$\mathcal{E} = \frac{1}{2}m\dot{x} + \frac{1}{2}kx^2$ The derivative is $$ \frac{d\mathcal{E}}{dt} = \frac{1}{2}m\ddot{x} + kx\dot{x} $$ but the solution is suppose to be $$ \frac{d\mathcal{E}}{dt} = \dot{x}(m\ddot{x} + kx). $$ How?

Hello! As of right now (10:13 PM), I've tried 9 combinations of points to solve this problem. It's a WebWork-based problem that's due in about an hour in a half. Any help would be very, very appreciated. Homework Statement I was given this equation: ##ln(2y) = 2xy## and was asked to find...

I've read about it before and now I'm trying to learn it myself from Woods 'Advanced Calculus' (as well as other resources like http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf) In the pdf, it says the method concerns integrals that depend on a parameter...now couldn't we...

Homework Statement The line segment AB lies on a diameter of a circle of radius 1, and the angle BAC is a right angle. Find the greatest possible value of the sum of the lengths of AB and AC. Homework Equations The Attempt at a Solution I have no idea what parameters to use...

Homework Statement x(t) = (t^2 -1) / (t^2 +1) y(t) = (2t) / (t^2 +1) at the point t=1 Homework Equations Line equation = y-y1 = m(x-x1) chan rule = (dy/dt) / (dx/dt) = dy/dx The Attempt at a Solution I find the y1 and x1 values by subing in t=1 to the x(t) and y(t)...

Homework Statement Find f^n for f(x) = Ln(2x+1) Can anyone point me in the right direction with how to get the nth derivative of the above function please, I just cannot seem to work this out! Thank you

Please see attached. I was looking for an explanation of the answer I have attached. Its been a little while and was just looking for the logic behind the differentiation shown for this problem. Its basically an optimization problem where I am looking for the minimum angle (theta) for the...

A rectangle of length x, where x varies, has a constant area of 48cm2. Express the perimeter, y in terms of x. Find the least possible value of x. my problem is not the maths part i.e. the differentiation, but the equation to get things moving. I really have no idea where to start. I drew a...

Homework Statement The function f(x,y,z) may be expressed in new coordinates as g(u,v,w). Prove this general result: The Attempt at a Solution df = (∂f/∂x)dx + (∂f/∂y)dy + (∂f/∂z)dz dg = (∂g/∂u)du + (∂g/∂v)dv + (∂g/∂w)dw df = dg since they are the same thing? but the...

Homework Statement I'm reviewing physics using Feynman's Lectures, and I'm finding that he frequently uses implicit differentiation in his lessons. This is unfortunate for me because I never got the hang of it beyond the simplest cases. I'm currently going through the proof that the...

Hi .. Use logarithmic differentiation to find the derivative can please check my answer and How I can know if the question want answer by using logarithmic differentiation or not ?

if a function f is differentiable of [0,2pi] can I integrate its derivative df on [-pi, pi]?

Homework Statement Use implicit differentiation to find dy/dx. Homework Equations xey - 10x + 3y = 0 The Attempt at a Solution = [xey + ey(y)'] - (10x)' + (3y)' = 0 = xey + ey(y') - 10 + 3(y') = 0 = y' (ey + 3) = 10 - xey = y' = 10 - xey/ ey + 3y However, my book says the answer: 10 -...

Homework Statement Use the quotient rule to differentiate y=(〖2x〗^4-3x)/(4x-1) Homework Equations y=(v du/dx-u dv/dx)/v^2 The Attempt at a Solution Please also find attached attempt as jpeg for clarity, and textbook supplied answer...

Homework Statement Consider the following equality: (\frac{∂S}{∂V})T = (\frac{∂P}{∂T})V If I rearrange the equality so that I write: (\frac{∂S}{∂P})? = (\frac{∂V}{∂T})? What variables will be constant in each side? I'm having some trouble in a few thermodynamics problems because...

Homework Statement Okay, the concept here is to use induction to prove that for n, (f1 x f2 x ... x fn-1 x fn)' = (f'1 x f2 x ... x fn) + (f1 x f'2 x ... x fn) + ... + (f1 x f2 x ... x f'n). 2. Homework Equations / 3. The Attempt at a Solution I solved the initial step, which was quite...

Homework Statement p(x)=vx^{n+1}+ux^{n}+1 Homework Equations 1) Find u and v so that 1 is a double root for p. 2) Conclude the quotient of p(x) over (x+1)^2. 3) For n=4 find u and v and find the quotient of p(x) over (x-1)^2. The Attempt at a Solution Can someone just tell me how to...

Homework Statement The equations ##2x^3y+yx^2+t^2=0##, ##x+6+t-1=0## implicitly define a curve $$f(t) = \begin{pmatrix} x(t)\\y(t) \end{pmatrix}$$ that satisfies ##f(1)=\begin{pmatrix} -1\\1 \end{pmatrix}.## Find the tangent line to the curve when ##t=1##. Homework Equations The...

Homework Statement Find the eigenvectors and eigenvalues of the differentiation map C1(R) -> C1(R) from the vector space of differentiable functions to itself. Homework Equations The Attempt at a Solution Hi, I'm not entirely sure how to go about this, because would the...

Homework Statement Find the derivative using logarithmic differentiation: y=(x+5)(x+9) The Attempt at a Solution lny=ln(x^2+14x+45) lny=(2x+14)/(x^2+14x+45) y'=(x^2+14x+45)((2x+14)/(x^2+14x+45))However, I know the derivative of the function is actually 2x+14. So I am wondering what is wrong...

Hello friends: My Question: A massive object cannot move at the speed of light. Photons can move at the speed of light because they are massless. However, since energy and mass are equivalent, due to Einstein's famous equation E^2=(m(c^2))^2+(pc)^2, mass is energy by a conversion...

Hi all I am trying to solve for an integral whose integrand is a derivative that has a change of variable inside of it. ∫ (dz/dx) * cos(θ) dθ between 0 and pi. I have a function for z(x), and also know the relation between of x and θ, but what I don't know is how to evaluate such...

Homework Statement I have a question. How in general would one differentiate a composite function like F(x,y,z)=2x^2-yz+xz^2 where x=2sint , y=t^2-t+1 , and z = 3e^-1 ? I want to find the value of dF/dt evaluated at t=0 and I don't know how. Can someone please walk me through this?Homework...

Equation: eysinx=x+xy I took the derivative of both sides. For the side with eysinx, I used the product rule and chain rule to get: ey*cosx + ey*sinx*y' For the side with x+xy, I used the sum and product rule to get 1+y+xy' So my resulting equation is: ey*cosx + ey*sinx*y'=1+y+xy', which...

Homework Statement Calculate ∂f/∂x and ∂f/∂y for the following function: yf^2 + sin(xy) = f The Attempt at a Solution I understand basic partial differentiation, but I have no idea how to approach the f incorporation on both sides of the equation nor what you would explicitly call this...

1. Homework Statement [/b] Find the derivative of the given function. Homework Equations Chain rule and logarithmic differentiation. The Attempt at a Solution See attached .gif. I was just wondering if this seemed correct? Thanks!

1. Given that y^{2}-2xy+x^{3}=0, find \frac{dy}{dx} 2. (no relevant equations other than the problem statement) 3. So, I solved it like this, \frac{dy}{dx}y^{2}-2xy+x^{3}=0 2y\frac{dy}{dx}-2+3x^{2}=0 Solving for dy/dx I got... \frac{dy}{dx}=\frac{-3x^{2}+2}{2y}...

Okay so, I am having trouble figuring out what exactly to do in implicit differentiation and usage of the chain rule. Like, I keep getting the wrong answer somehow. See, from what I understand you have to find the derivative of both sides then use the chain rule or something and then solve for...

Hello, I have recently started a little implicit differentiation and I have seen DEs before but I know that I still need to work on my differentiation and integration a little more before I am ready to tackle those. Anyway, I wish to ask, what distinguishes implicit differentiation from a...

Homework Statement I need to use implicit differentiation to find the derivative of y=sin(x+y). Homework Equations The Attempt at a Solution This is what I did: y=sin(x+y) y'=(sin(x+y))' y'=(1+y')(cos(x+y)) (by the chain rule) Now, what do I do? Is this correct...

Homework Statement Show that B = {x2 −1,2x2 +x−3,3x2 +x} is a basis for P2(R). Show that the differentiation map D : P2(R) → P2(R) is a linear transformation. Finally, find the following matrix representations of D: DSt←St, DSt←B and DB←B. Homework Equations The Attempt at a...

Homework Statement solve ∫log(1+acosx) by differentiation under integral sign (limits are 0 to ∏) Homework Equations The Attempt at a Solution =∫(1/1+acosx)cosxdx(by leibinitz by differentiating partially WRT a. Then how do I proceed,can anyone show me all the steps of...

1. Water flows out of a cylindrical tank under gravity via a tap, the height h(t) of the water column above the tap satisfies the differential equation in the form dh/dt = -2k√h where k is some positive constant. The water column has a height initially of 25m. The tap is turned on and...

Homework Statement Find the derivatives at an arbitrary point x in the domain of the following functions f_i: D_i → ℝ, where for 1 ≤ i ≤ 6 the domain D_i is the maximal subset of ℝ on which the mapping is defined - you don't have to determine the domains. Homework Equations a) f_1 (a) =...

Homework Statement The height "s" at time of a silver dollar dropped from a building is given by s(t) = -16t^2 + 1350, where "s" is measured in feet and "t" is measured in seconds [s'(t) = -32t] a) Find the average velocity on the interval [1,2]. ( I ALREADY SOLVED) b) Find the instantaneous...

1. y = sinxy Homework Equations 3. this was my attempt d/dx = (cosxy)(sinxy(d\dx))+(xy(d/dx) im getting stuck. i don't think I am starting it right. any suggestions.

Homework Statement Find the derivative of y=(x^2)^sinx; using the chain rule. Homework Equations No other relevant equations. The Attempt at a Solution I attempted to apply the Chain rule: dy/dx = dy/du X du/dx Subbing u for x^2, which made y = u^sinx I ended up with...

Homework Statement Find \frac{dy}{dx}. y=\sin^{-1}(2x\sqrt{1-x^2}), \frac{-1}{\sqrt{2}}<x<\frac{1}{\sqrt{2}}Homework Equations The Attempt at a Solution I started with substituting x=sinθ. The expression simplifies to y=\sin^{-1}(\sin(2θ)) which is equal to y=2θ. Substituting back the value of...

Hi, Just a question on an example in a maths textbook. See attached image for question below. So, I understand that if you set u=sin(x) and v=e^-cos(x) f'(x)=u'.v + u.v' But I'm stuck looking at e^-cos(x), could it also be classified e^(w)? Also, the second step in differentiating...

Folks, I am stuck on an example which is partial differenting a functional with indicial notation The functional ##\displaystyle I(c_1,c_2,...c_N)=\frac{1}{2} \int_0^1 \left [ \left (\sum\limits_{j=1}^N c_j \frac{d \phi_j}{dx}\right )^2-\left(\sum\limits_{j=1}^N c_j \phi_j\right)^2+2x^2...

I've got some maths homework to do over the summer before I go back to uni and there's this stupid question on there which is one of those 'so basic I don't know it' kind of questions, so here goes. Homework Statement What is LimΔx→0\frac{y(x+Δx) - y(x)}{Δx} ? The Attempt at a Solution...

Hi there! If one would want to prove that the indefined integral : \int[f(x)+g(x)]dx = \int f(x)dx + \int g(x)dx. Would this be apropriate: A(x) = \int[f(x)+g(x)]dx; B(x) = \int f(x)dx; C(x) = \int g(x)dx. And since the primitive of a fuction is another fuction whose derivative...

what exactly does Δy/Δx mean. for instance i know that when y=x2 Δy/Δx=2x but what does Δx/Δy equal? also why is the derivative always Δx/Δy? also what does Δx by itself mean for instance if y=x2 what is Δx i appreciate any and all answers thanks:smile:

Homework Statement Find the velocity and acceleration of a particle with the given position function: r(t)=<2cos t, 3t, 2sin t/t+1> The Attempt at a Solution v(t)=r'(t) dt= <-2sin t, 3, (2cos t/t+1) - (2sin t/(t+1)2)> a(t)=v'(t) dt = <-2cos t, 0, (4sint t/(t+1)3-(2sint t/(t+1)-(4...

Homework Statement This question is about entropy of magnetic salts. I got up to the point of finding H1, the final applied field. The Attempt at a Solution But instead of doing integration I used this: dS = (∂S/∂H)*dH = (M0/4α)(ln 4)2 I removed the negative...

Homework Statement I'm try to implicitly differentiate the function: xlny+√y=lnx The Attempt at a Solution And I got to the stage where I have: dy/dx = (1/x-lny)/(x/y+1/(2*√y)) which is where I have no idea on how to clean this up. Could someone please explain to me how to simplify a...

Hi everyone, I need help with a derivation I'm working on, it is the differentiation of the norm of the gradient of function F(x,y,z): \frac{∂}{∂F}(|∇F|^{α}) The part of \frac{∂}{∂F}(\frac{∂F}{∂x}) is bit confusing. (The answer with α=1 is div(\frac{∇F}{|∇F|}), where div stands for...