Derivatives Definition and 1000 Threads

I've hit a snag in my studies, namely something my book labels "Corollary 10.1": [i]Are there any other functions with the same derivative as x^2+2=2x? You should quickly come up with several: x^2+3 and [itex x^2-4[/itex] for instance. In fact, d/dx[x^2+c]=2x for any constant c. Are there...

The problem I'm curious about is this: \frac{\partial}{\partial r}(\frac{\partial r}{\partial θ}) I found that the answer is zero using WolframAlpha, but obviously I won't have that on a future test xD. Can someone please explain to me how to think about the derivative above? How can I look...

Homework Statement let u=f(x,y) , x=x(s,t), y=y(s,t) and u,x,y##\in C^2## find: ##\frac{\partial^2u}{\partial s^2}, \frac{\partial^2u}{\partial t^2}, \frac{\partial^2u}{\partial t \partial s}## as a function of the partial derivatives of f. i'm not sure I'm using the chain rules...

Suppose we have a torsion free connection. Does anyone here know of a slick way to prove that covariant derivatives satisfy the Jacobi identity? I.e. that $$([\nabla_X,[\nabla_Y,\nabla_Z]] + [\nabla_Z,[\nabla_X,\nabla_Y]] +[\nabla_Y,[\nabla_Z,\nabla_X]])V = 0$$ without going into...

Se a function f(x(t, s), y(t, s)) have as derivative with respect to t: \frac{df}{dt}=\frac{df}{dx} \frac{dx}{dt}+\frac{df}{dy} \frac{dy}{dt} And, with respect to s: \frac{df}{ds}=\frac{df}{dx} \frac{dx}{ds}+\frac{df}{dy} \frac{dy}{ds} But, how will be the derivative with respect to...

Homework Statement This is an Optimization Problem, find the maximum value. P(R)=(E^2*R)/(R+r)^2 Homework Equations P'(R)=? The Attempt at a Solution I have the solution to this problem, and I can solve it, I just don't understand some parts. I tend to think that the...

First a little warm up problem. Suppose g:\mathbb{R}^N\to\mathbb{C} is some fixed function, and we want to find f:\mathbb{R}^N\to\mathbb{C} such that g(x) = u\cdot\nabla_x f(x) holds, where u\in\mathbb{R}^N is some constant. The problem is not extremely difficult, and after some work...

I determined the equilibrium point of a spring by setting the potential energy function U(r) equal to zero and solving for r. But I just looked at the guided solution, and they took the derivative of U(r) first, then solved for r. Is my approach correct? Can we solve for the equilibrium...

Consider a function f(x), such that for all points x0 in the domain, the nth derivative of f evaluated that x0 is less than the n+1th derivative of f evaluated at x0. A quick example is f(x) = e^(ax) where a > 1, what others are there (not including just changing e to something else)? Is...

Greetings, I want to become more fluent using functional derivatives. Does anyone have a link to sets of problems involving functional derivatives or anything like that (e.g., a worksheet from a class where they were used or something)? The lengthier the better, and ideally the solutions...

Sorry, wasn't sure how to describe the problem in the title. Homework Statement Okay, this problem has really been bugging me for a while. It's a question that I thought of when I was daydreaming in class, but now I can't stop thinking about it. I've been going crazy for days Here is the...

Homework Statement Find the second derivative of $$9x^2+y^2=9$$ Homework Equations Chain rule The Attempt at a Solution I find the first derivative first. $$18x+2y\frac{dy}{dx}=0$$ $$\frac{dy}{dx}=-9\frac{x}{y}$$ I then find the second derivative...

Homework Statement Find the derivative of $$y=cos(\frac{1-e^{2x}}{1+e^{2x}})$$ Homework Equations Chain rule The Attempt at a Solution $$y=cosu$$ $$\frac{dy}{du}=-sinu$$ $$u=\frac{1-e^{2x}}{1+e^{2x}}$$ $$ \frac{du}{dx}=(1-e^{2x})(-(1+e^{2x})^{-2})+(1+e^{2x})^{-1}(-2e^{2x})$$...

Homework Statement Find the derivative of y=cos(a3+x3) Homework Equations Chain rule The Attempt at a Solution y=cosu \frac{dy}{du} = -sinu u=a3+x3 \frac{du}{dx} = 3a2+3x2 \frac{dy}{dx} = -3sin(a3+x3)(a2+x2). The answer is supposed to be -3x2sin(a3+x3). What did...

Homework Statement Find the derivative of y=xe-kx Homework Equations Chain rule The Attempt at a Solution y = xeu \frac{dy}{du} = xeu+eu u = -kx \frac{du}{dx} = -k \frac{dy}{dx} = (xe-kx+e-kx)(-k) = e-kx(x+1)(-k) = e-kx(-kx-k) The answer is e-kx(-kx+1)...

1. Find the derivative of y=e\sqrt{x} Homework Equations Chain rule The Attempt at a Solution y=eu \frac{dy}{du}= ueu-1 u=\sqrt{x} \frac{du}{dx}= \frac{1}{2}x-1/2 \frac{dy}{dx}= \sqrt{x}e\sqrt{x}-1 × \frac{1}{2}x-1/2 = \sqrt{x} \frac{e^\sqrt{x}}{e} ×...

1. A ladder 10 ft long rests against a vertical wall. Let θ be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x change with respect to θ when...

Homework Statement http://i3.minus.com/j7uTkNLAl2aBy.png Homework Equations Extrema occur at critical points; critical points are either where the first derivative fails to exist or equals 0. Horizontal tangent lines occur where the first derivative is 0. Points of inflections occur...

The standard definition of the lie derivative of X along Y is just $$(*) \mathcal{L}_YX = \lim_{t\to 0} \frac{X_{\phi(t)} - \phi_{t*}X}{t}$$ where ##\phi_t## is the flow generated by Y. I.e. the limit of the difference between a pushforward of X along Y and X evaluated at a point...

Looking through this matrix approach to the quantum harmonic oscillator, http://blogs.physics.unsw.edu.au/jcb/wp-content/uploads/2011/08/Oscillator.pdf especially the equations m \hat{ \ddot { x } } = \hat { \dot {p} } = \frac {i}{\hbar} [ \hat {H} , \hat {p} ] I'm getting the impression...

Homework Statement Show that y= xex satisfies A(d2y)/dx2 + B(dy/dx) + Cy = 0 for suitably chosen values of the constants A, B, and C. Homework Equations Y=xex The Attempt at a Solution Please see the attachment. I get to a point where I need to find the value of A, B...

## \vec{r}=\rho \cos \varphi \vec{i}+\rho \sin \varphi \vec{j}+z\vec{k} ## we get \vec{e}_{\rho}=\frac{\frac{\partial \vec{r}}{\partial \rho}}{|\frac{\partial \vec{r}}{\partial \rho}|} \vec{e}_{\varphi}=\frac{\frac{\partial \vec{r}}{\partial \varphi}}{|\frac{\partial \vec{r}}{\partial...

Homework Statement I need to prove that directional derivatives do not commute. Homework Equations Thus, I need to show that: (\vec{A} \cdot \nabla)(\vec{B} \cdot \nabla f) - (\vec{B} \cdot \nabla)(\vec{A} \cdot \nabla f) = (\vec{A} \cdot \nabla \vec{B} - \vec{B} \cdot \nabla...

Hi all, This is my first post here. As a bit of background I have a keen interest in mathematics, but I'm not necessarily strong in it... Today I was working with a dataset that models difference in elevation values (dH) over time (x). I have approximately 20 samples, of which each dH value...

Homework Statement I failed my math exam last year (first year of college) because I basically had no math classes during my last two years of high school. So I would like to learn the specific materials to be able to solve these problems. So I guess this isn't a traditional question, but I do...

Homework Statement A point particle has a position vector r⃗ (t) as a function of time t, given by r(t)=(1−t2)x^−2t(t+5)y^+8(t+2)z^. where distances are in meters, and time t is in seconds. Now, let t=t1= 23 s. What is the (smaller) angle between the velocity vector at time t1 and the z^...

Is the grad(\frac{\partial f}{\partial t}) the same as \frac{\partial}{\partial t}(gradf)? Thank you.

The question is: a) Find explicit expressions for an ideal gas for the partial derivatives: (∂P/T)T, (∂V/∂T)P and (∂T/∂P)V b) use the results from a) to evaluate the product (∂P/V)T*(∂V/∂T)P*(∂T/∂P)V c) Express the definitions of V(T,P) KT(T,P)an BT(T,V) in terms of the indicated independent...

I don't understand this concept very well. So suppose I want to find the derivative of the function f(x)=2. is the answer just zero? what if I want to find f'(x)=2x, or f'(x)=2x3? please explain how to get the answer step by step, is there some equation I use or something? I just don't get it!

Homework Statement Let ##f(x)## and ##g(x)## be two differentiable function in R and f(2)=8, g(2)=0, f(4)=10 and g(4)=8 then A)##g'(x)>4f'(x) \forall \, x \, \in (2,4)## B)##3g'(x)=4f'(x) \, \text{for at least one} \, x \, \in (2,4)## C)##g(x)>f(x) \forall \, x \, \in (2,4)##...

Homework Statement Show that if f^{(n)}(x_0) and g^{(n)}(x_0) exist and \lim_{x \rightarrow x_0} \frac{f(x)-g(x)}{(x-x_0)^n} = 0 then f^{(r)}(x_0) = g^{(r)}(x_0), 0 \leq r \leq n . Homework Equations If f is differentiable then \lim_{x \rightarrow x_0}\frac{f(x)-T_n(x)}{(x-x_0)^n}=0 ...

I am given Z = f (x, y), where x= r cosθ and y=r sinθ I found ∂z/∂r = ∂z/∂x ∂x/∂r + ∂z/∂y ∂y/∂r = (cos θ) ∂z/∂x + (sin θ) ∂z/∂y and ∂z/∂θ = ∂z/∂x ∂x/∂θ + ∂z/∂y ∂y/∂θ= (-r sin θ) ∂z/∂x + (r cos θ) ∂z/∂y I need to show that ∂z/∂x = cos θ ∂z/∂r - 1/r * sin θ ∂z/∂θ and ∂z/∂y = sin...

Homework Statement how to find nth derivatives cos^12x and a-x/a+x Homework Equations The Attempt at a Solution

Hi, I am sort of hung up with a particular step in a derivation, and this has caused me to ponder a few properties of partial derivatives. As a result, I believe I may be correct for the wrong reasons. For this example, the starting term is (\frac{\partial}{\partial x}\frac{\partial...

I'm going to talk with someone at a local university tomorrow to see if I can get out of AP Calculus. Essentially, I would like to be prepared for our meeting tomorrow. I'm good with integrals, so that shouldn't be a problem. However, I'm not quite as confident with derivatives, limits, and...

Is there a general formula for (total) derivatives of functions of the form f(xy(x)+z(x)? I tried the most simple function of that form f(xy(x)+z(x))=xy(x)+z(x) and the formula I got was \frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} \frac{dy}{dx}+\frac{\partial f}{\partial...

I am having a little trouble remembering the rules with derivatives. $\frac{a}{1-r}$ I know that it should be (derivative of the top*bottom - top*derivative bottom) / (bottom squared). $\frac{d}{dr}\frac{a}{1-r}$ I tried this got the answer wrong, and looked up how to do this and they showed...

The volume of a sphere with radius r is v = \frac{4}{3}\pi r ^{3} It makes sense that its derivative with respect to radius is the surface area of the sphere. \frac{dv}{dr} = a = 4\pi r ^{2} The volume of a cube with side length n is n^{3} The derivative of this is just 3n^(2)...

Homework Statement Use the definition of a derivative to compute the derivative of the function at the given value. P(t)=t^3-4, t=-2 Homework Equations Please help me solve it in this form: f(a+h)-f(a)/h lim h→0 The Attempt at a Solution...

Is there a standard notation for partial derivatives that uses indexes instead of letters to denote ideas such as the 3 rd partial derivative with respect to the the 2nd argument of a function? As soon as a symbol gets superscripts and subscripts like \partial_{2,1}^{3,1} \ f the spectre of...

I don't understand what the derivatives really mean?I know that they are the slope of the tangents drawn to a function.But see for example we have a function f(x)=x2 The derivative of this gives us '2x'. But what does '2x' mean?If i draw a graph of f(x)=2x what does it give me?what should i...

Hello, I need to find a two-arguments function u(x,y) which satisfies six constraints on its derivatives. x and y are quantities so always positive. 1&2: On the first derivatives: du/dx>0 for all x & du/dy>0 for all y (so u is increasing in x and y) 3&4: On the second derivatives...

I am a little confused about when to use the quotient rule. When you have one function over another function, and are taking the derivative, are you required to use this technique? I thought you were, but then I was watching this video on Khan Academy...

Hi everyone Homework Statement I want to find the multipole expansion of \Phi(\vec r)= \frac {1}{4\pi \epsilon_0} \int d^3 r' \frac {\rho(\vec r')}{|\vec r -\vec r'|} Homework Equations Taylor series The Attempt at a Solution My attempt at a solution was to use the Taylor series. I...

I was doing a proof on why the derivative of an even function is odd and vice versa. Now, the way I did the problem was by using the chain rule to rewrite the derivative of f(-x), and the proof worked out perfectly fine. But I had a thought that I can't quite wrap my around, and I think it's...

I understand derivatives and I am not trying to be like a stickler or anything, but before manipulating the equation to arrive at a form where we can find a real answer for a derivative, we are left with [f(x+h)-f(x)]/h (where h is delta x I guess as most people write it). Before evaluating...

Partial Derivatives Hi all I was wondering if anyone could help me with this problem. I have a triangle that has a = 13.5m, b = 24.6m c, and theta = 105.6 degrees. Can someone remind me of what the cosine rule is? Also (my question is here) From the cosine rule i need to find: the...

I'm trying to figure out the general solution to the integral ##\int \frac{d^ny}{dx^n} \, dy##, where n is a positive integer (Meaning no fractional calculus. Keeping things simple.). So far, I have been working with individual cases to see if I can establish a general pattern and then try a...

Homework Statement Find f'(x): (abs(((x^2)*((3x+2) ^(1/3)))/((2x-3)^3)) <- Not sure why it''s not showing up but the 1/3 is an exponent to just the (3x+2).Homework Equations Product Rule and Quotient Rule for DifferentiatingThe Attempt at a Solution So I thought I should split it into two...

Homework Statement Suppose that f' (2) = 3. Find the limit as x approaches 2 of [f(x)−f(2)]/[sqrt(x) - sqrt(2)] Answer: 6*sqrt 2 Homework Equations The Attempt at a Solution f'(x) = lim h->0 = [f(a +h) - f(a)]/h = slope [f(x)-f(2)]/ x-2 a = 2 i would think that the limit =...