Derivatives Definition and 1000 Threads

Homework Statement 1. Is (∂P/∂x)(∂x/∂P) = 1? I realized that's not true, but I'm not sure why.2. Say we have an equation PV = T*exp(VT) The question wanted to find (∂P/∂V), (∂V/∂T) and (∂T/∂P) and show that product of all 3 = -1.The Attempt at a SolutionI tried moving the variables about...

Hello, I was pondering acceleration's relation to velocity, when I came across a problem that is more of an issue of differentiation. Given our variables v=velocity a=acceleration α=slope of 'a' t=time (at1→t2)=Total acceleration from t1 to t2 (αt1→t2)=Total slope of 'a' from t1 to t2...

Homework Statement Hi I have a technical question regarding the following paper: http://arxiv.org/pdf/quant-ph/0602170v1.pdf In it they derive an equation for \left\langle a \right\rangle (equation #5) from the master equation (equation #2). My question is how they do this. Here is what...

I just got to a point in multivariable calculus where I realize I can solve problems in assignments and tests but have no actual idea of what I'm doing. So I started thinking about stuff and came up with a few questions: 1. Is picturing the derivative as the slope of the tangent line to a...

Homework Statement From step 1 to step 2, what do they mean by "Taking the weighted sum of the two squares " ? I tried and expanded everything in step 2 and it ends up as the same as step 1 (as expected), The Attempt at a Solution I tried looking up "weighted sum" and "...

Is there any clear explanation as to why exactly derivatives and integrals cancel each other [other than the integral is the anti-derivative]? To my understanding the derivative gives the slope of a curve at given point whereas the the integral finds the area under the curve. How are these...

Hi! I'm currently re-reading Griffiths introductory QM book and plan to do most of the exercises. I got stuck on one problem and had to look for some hints and found two solutions that both claim that: \int_{-\infty} ^{\infty} \frac{\partial }{\partial x}\left[ \frac{\partial \Psi...

Here's the problem: Find the directional derivative of the function $V(x,y) = x^3 -3xy +4y^2$ at the point $P(2,1)$ in the direction of the unit vector $\vec{u}$ given by the angle $\theta = \pi /6$. So I found the gradient of the function $$\nabla V(x,y) = (3x^2 -3y, 8y -3x),$$ and then at...

Homework Statement http://gyazo.com/d875970963124b7d4a64acc887f168fa The Attempt at a Solution I don't understand the part where they take the derivative of a diff eq to find the higher derivatives can somebody explain it? Mainly the answer where they get y''=(y'')'=(1-y^2)'=-2yy'(this...

I'll just make one thread for all the help I'll need with derivatives so I don't clutter up this forum. Homework Statement Find the derivative of y = sqrt(x)(x - 1).Homework Equations Wolfram Alpha gets this: http://www.wolframalpha.com/input/?i=derivative+y+%3D+sqrt%28x%29%28x+-+1%29 I got...

Hi all! I'm sorry if this question has been already asked in another post... I'm studying the path integrals formalism in QED. I'm dealing with the functional generator for fermionic fields. My question is: The generating functional is: $$Z_0=e^{-i\int{d^4xd^4y \bar{J}(x)S(x-y)J(y)}}$$...

Homework Statement Consider the surface and point given below:- Surface: f(x,y)= 4-x2-2y2 Point: P(1,1,1) a) Find the gradient of f. b) Let C' be the path of steepest descent on the surface beginning at P and let C be the projection of C' on the xy-plane. Find an equation of C in the...

Hi I thought of something today coming home from school: If I have two arbitrary real functions f(x) and g(x) and I know that \frac{df(x)}{x} = \frac{dg(x)}{dx} Does this imply that f(x)=g(x)? Niles.

In Spivak's Calculus, there is a theorem relating the derivative of the limit of the sequence {fn} with the limit of the sequence {fn'}. What I don't like about the theorem is the huge amount of assumptions required: " Suppose that {fn} is a sequence of functions which are differentiable on...

Homework Statement Find a differential of second order of a function u=f(x,y) with continuous partial derivatives up to third order at least.Hint: Take a look at du as a function of the variables x, y, dx, dy: du= F(x,y,dx,dy)=u_xdx +u_ydy. Homework Equations The Attempt at a Solution I'll be...

Homework Statement I have got a question concerning the following function: f(x,y)=\log\left(x^2+y^2\right) Partial derivatives are: \frac{\partial^2f}{\partial x^2}=\frac{y^2-x^2}{\left(x^2+y^2\right)^2} and \frac{\partial^2f}{\partial y^2}=\frac{x^2-y^2}{\left(x^2+y^2\right)^2} The...

Homework Statement Let f be a polynomial function of degree n such that f(x) ≥ 0 for all x (note that n must be even). Prove that f + f' +f'' + f''' + ... + f^(n) ≥ 0. Homework Equations I believe that is all - the derivative of some term in the polynomial ax^n is anx^(n-1). The...

Hello Everyone! This has been confusing me a lot: consider a function $f(x) = x^2 + 2x + 3$. Now, $\frac{\partial f}{\partial x} = 2x + 2$. Now, someone tells me that $y = x^2$. What is $\frac{\partial f}{\partial x}$ now?

Why is partial derivative with respect to time used in the continuity equation, \frac{\partial \rho}{\partial t} = - \nabla \vec{j} If this equation is really derived from the equation, \frac{dq}{dt} = - \int\int \vec{j} \cdot d\vec{a} Then should it be a total derivative with...

I luckily went on to Newton's laws which I already know. I was stuck with a fantastic question ( I have been seeing the Newton's laws from my childhood, but I didn't notice these sort of observation ) . I was seeing the Force and Momentum. Force is based upon acceleration and Momentum on...

In the ordinary least squares procedure I have obtained an expression for the sum of squared residuals, S, and then took the partial derivatives of it wrt β0 and β1. Help me to condense it into the matrix, -2X'y + 2X'Xb. ∂S/∂β0 = -2y1x11 + 2x11(β0x11 + β1x12) + ... + -2ynxn1 + 2xn1(β0xn1 +...

Help Derivatives ASAP -- maximizing sunlight through a window 3. The amount of daylight a particular location on Earth receives on a given day of the year can be modeled by a sinusoidal function. The amount of daylight that Windsor, Ontario will experience in 2007 can be modeled by the function...

Derivatives Question -- temperature as a funtion of time 1. When a certain object is placed in an oven at 540°C, its temperature T(t) rises according to the equation T(t) = 540(1 – e–0.1t), where tis the elapsed time (in minutes). What is the temperature after 10 minutes and how quickly is it...

Dear all, I have a confusion about partial derivatives. Say I have a function as y=f(x,t) and we know that x=g(t) 1. Does it make sense to talk about partial derivatives like \frac{\partial y}{\partial x} and \frac{\partial y}{\partial t} ? I doubt, because the definition of...

Homework Statement Hi I've been giving the following problem: We have a differentiable function f: [a,b] \rightarrow \mathbb{R} with f'(a) < 0 en f'(b) > 0. Let c \in \mathbb{R} such that f'(a) < c. Show that there exists a \delta >0 such that for every x \in ]a, a + \delta[ the following...

Why is it that the constant e is defined as being the unique case where the limit as h goes to 0 of (e^h - 1)/h = 1? I mean every exponential function like a^x equals 1 when x equals 0, right? So would it be fair to say that (a^h) approaches 1 as h approaches zero? And that (a^h - 1) approaches...

The total derivative of the function z=f(x,y) with respect to x is: dz/dx = ∂z/∂x + (∂z/∂y)(dy/dx) The way i see this is that the total derivative, dz/dx, gives the rate of change of z with x, allowing y to vary with x at the rate dy/dx. I don't know if this is right. The directional...

Homework Statement Here is the problem: http://dl.dropbox.com/u/64325990/MATH%20253/help.PNG The Attempt at a Solution http://dl.dropbox.com/u/64325990/Photobook/Photo%202012-05-24%209%2037%2028%20PM.jpg This seems to be wrong... Since I have fx and fy which I cannot cancel out. Why...

Homework Statement Here is the problem with the solution: http://dl.dropbox.com/u/64325990/MATH%20253/Capture.PNG What I don't understand is circled in red, how did they combine dxdy with dydx? Is it with Clairaut's theorem? If it is can someone explain how it works in this case because...

Homework Statement Ok, I'm trying to understand the proof for derivatives. I understand most of it, but there is one step that I cannot understand. lim x-> 0 [xm - am]/[xn - an = (m/n)am-n The Attempt at a Solution I don't see how those are equal. The best I can do is...

Hi, I was hoping someone could help me out with my homework set. I have done a lot of the questions, and it would help if someone could tell me if I have done them correctly. Thanks! :) Q1: Find first derivatives for the following functions (a)g(s,t)=sin(st^3)...

Directional and partial derivatives help please! I have read that the partial derivative of a function z=f(x,y) :∂z/∂x, ∂z/∂y at the point (xo,yo,zo)are just the tangent lines at (xo,yo,zo) along the planes y=yo and x=xo. Directional derivatives were explained to be derivatives at a particular...

1. A 1000 L tank is draining such that the volume V of water remaining in the tank after t minutes is v=1000(1-t/60)2. Find the rate at which the water is flowing out of the tank after 10 min. Calculate dV/dt using chain rule u = 1 - t/60: u = 1 - t/60 ==> du/dt = -1/60 V = 1000u^2 ==> dV/du =...

f=9-x^2-y^2 and u=i-j The directional derivative comes out to be Du f(x,y)=-sqrt(2)+sqrt(2) I'm going to find the directional derivative and could someone kindly point out the mistake because I am getting a different answer and it's important I understand how to do this question: Du...

I was searching for the proof of \frac{d}{dx} e^x = e^x. and I found one in yahoo knowledge saying that \frac{d}{dx} e^x = \lim_{Δx\to 0} \frac {e^x(e^{Δx}-1)} {Δx} = \lim_{Δx\to 0} \frac {e^x [\lim_{n\to\infty} (1+ \frac{1}{n})^{n(Δx)}-1]} {Δx} Let h= \frac {1}{n} , So that n = \frac...

Homework Statement Demonstrate that C_{Y,N}=\left ( \frac{ \partial H}{\partial T } \right ) _{Y,N} where H is the enthalpy and Y is an intensive variable. Homework Equations (1) C_{Y,N}=\frac{T}{N} \left ( \frac{ \partial S}{\partial T } \right ) _{Y,N} (2) T= \left ( \frac{ \partial...

Homework Statement Find the derivative of f(x): f(x)= 1/ ((ln(x)^2)) Homework Equations f(x)= ln(x) f'(x)= 1/((ln(x)) The Attempt at a Solution Dx(1/ln(x)^2) = Dx((ln(x))^-2)= -2*(ln(x)^-3) * Dx(ln(x)) = -2*(ln(x)^-3) * 1/x = -2/(x*ln(x)^3) Are these the correct...

I'm not sure if it's OK to post this question here or not, the Calculus and Beyond section doesn't really look very heavily proof oriented. I'm trying to prove that if continuous complex valued function f(z) is such that the directional derivatives(using numbers with unit length) preserve...

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I'm struggling to understand what the derivative of an attitude quaternion really is and how to use it. I need it to solve a problem relating to a rotating frame of reference relative an inertial frame. The information I have is a vector of Euler angular velocities (i.e for roll \phi, pitch...

f(x,y)=\sqrt{|xy|} Do the partial derivatives of f exist at x=0, y=0?

There is already an article on physics forums that kind of addresses my issue: https://www.physicsforums.com/showthread.php?t=107516 and I'm not really satisfied with the wikipedia article. I am generally confused on what the derivative should be. I'm familiar with Jacobian matrices but am...

Can this problem be solved? I made up the problem myself so I am not sure a solution exists. It is known that: h(0) = 0 h'(0) < 0 h'' > 0 Prove that there exists a value c > 0 such that h(c)=0 It makes sense visually. I have tried applying the MVDT/IVT in various ways, but...

Homework Statement If h(x):=0 for x<0 and h(x):=1 for x≥0. Prove there dne f:ℝ→ℝ such that f'(x) = h(x) for all x in ℝ. Give examples of two functions not differing by a constant whose derivatives equal h(x) for all x ≠ 0 Homework Equations The Attempt at a Solution I don't know...

For a function such as w=5xy/z How would you find the partial derivative of w with respect to y or z? I've tried using basic logarithmic differentiation, but can't arrive at the correct answer. For reference, the correct answer is wy=5*(xy/z/z)*ln(x)

1. In the Khan academy video I watched on partial derivatives, I understand absolutely everything except for the last 20 seconds which confused me. http://www.youtube.com/watch?v=1CMDS4-PKKQ Using the formula: Z = x² + xy + y² @z/@x = 2x +y x=0.2, y=0.3 2(.2) + .3 = .7 What...

Homework Statement A bush-walker is climbing a mountain, of which the equation is h \left( x,y \right) =400-{\frac {1}{10000}}\,{x}^{2}-{\frac {1}{2500} }\,{y}^{2} The x-axis points East, and the y-axis points North. The bush-walker is at a point P, 1600 metres West, and 400 metres South of...

Quick question. This is kind of embarrassing actually. Suppose I have functions x(t,s) and y(t,s) (say they're parametric equations of a surface of something) and I want to know what dy/dx is. Specifically, I have x and y in terms of the parameters, which are kind of complicated functions, and I...

Homework Statement Find the approximate percentage changes in the given function y = f(x) that will result from an increase of 2% y = x2 Homework Equations The Attempt at a Solution dy/dt = dx/dt * d/dx * x2 dy/dt = dx/dt * 2x dy/dt = 2/100 * 2x dy/dt = 4x% ? I don't know if I...

Homework Statement Hi. I am attempting to get the Euler-Lagrange equations of motion for the following Lagrangian: L(ψ^{μ}) = -\frac{1}{2} ∂_{μ} ψ^{\nu} ∂^{μ} ψ_{\nu} + \frac{1}{2} ∂_{μ} ψ^{\mu} ∂_{\nu} ψ^{\nu} + \frac{m^{2}}{2} ψ_{\nu} ψ^{\nu} Homework Equations So, I want to get...