Differentiation Definition and 1000 Threads

Defining the Hermite Polynomials by: \mathscr{H}_n(x) = (-1)^n e^{x^2}\, \frac{d^n}{dx^n} \Bigg\{ e^{-x^2}\Bigg\}Show that \mathscr{H}_5(x) = 32 x^5-160 x^3+120 x

Homework Statement Proof the following: \frac{\text{d}\boldsymbol\{\mathbf{I}\boldsymbol\}}{\text{d}t} \, \boldsymbol\omega = \boldsymbol\omega \times (\boldsymbol\{\mathbf{I}\boldsymbol\}\,\boldsymbol\omega) where \boldsymbol\{\mathbf{I}\boldsymbol\} is a tensor...

Homework Statement if z = x2 + 2y2 , x = r cos θ , y = r sin θ , find the partial derivative \left(\frac{\partial z}{\partial \theta}\right)_{x} Homework Equations z = x2 + 2y2 x = r cos θ y = r sin θ The Attempt at a Solution The textbook says that the equation should be...

Homework Statement Differentiate using implicit differentiation y^2sin(x) Homework Equations I know you need the chain rule and the product rule to solve this The Attempt at a Solution So, it would be: 2yy' + y^2cos(x) Is that correct?

Homework Statement Find the gradient of the tangent at x on the following curve ##y=3x^2## Homework Equations $$\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$The Attempt at a Solution I know that it's ##6x##. $$\frac{3(x+\Delta x)^2-3x^2}{\Delta x}$$ $$=\frac{3x^2+6x\Delta x...

Homework Statement A circuit consists of 230V supply, a switch, a 2mH inductor and a 12k ohm resistor in series When the switch is closed at time t=o, a current i begins to flow in the circuit: The current is modeled by the following equation: i= v/r (1-e^-Rt/L) Determine the...

If a repeated integral can be expressed how an unique integral: https://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration So is possible express the nth derivative with an unique differentiation?

Hi guys, I'm not sure where to put this question, so I'll just put it here. If a mod knows of a better place, just point me to it, thanks. I'm looking at the functional differentiation equation: $$\left.\frac{dF[f+\tau h]}{d\tau}\right|_{\tau=0}\equiv \int\frac{\delta F[f]}{\delta...

Hi all, I'm working on some QFT and I've run into a stupid problem. I can't figure out why my two methods for evaluating i\gamma^\mu \partial_\mu \exp(-i p \cdot x) don't agree. I'm using the Minkowski metric g_{\mu\nu} = diag(+,-,-,-) and I'm using \partial_\mu =...

Hi guys, I'm trying to study the functional approach to quantization in QFT. The QFT books seem to often "sweep things under the rug" and not be too rigorous when it comes to issues like integral convergence, and the like. So I was wondering if there was a more mathematically rigorous...

When I take the differential of y wrt t (being t a parameter (time)) I get the velocity of the y-coordinate, if take the second differential of y wrt t, thus I get the aceleration of the y-coordinate... ok! But what means to differentiate the y-coordinate wrt x-coordinate, or wrt y, or then...

Homework Statement Find the derivative of: x+xy=y^2 Homework Equations So I know you have to differentiate it, and it would be: 1+xyy'=2yy' The Attempt at a Solution Moving the terms with y' to one side: 1+xyy'-2yy'=0 xyy'-2yy'=-1 Factoring out y'...

Problem: Let $g(x)$ be twice differentiable function satisfying $g(0)=0$, $g(1)=1$. Then, which of the following is/are correct? A) there exist distinct $C_1,C_2\in (0,1)$ such that $g'(C_1)+g'(C_2)=2$. B) there will be atleast one $C$ such that $g'(C)=1$ for $C\in (0,1)$ C) there will be...

Homework Statement Show that if the vector function r(t) is continuously differentiable for t on an interval I and |r(t)| = c, a constant for all t \in I, then r'(t) is orthorgonal to r(t) for all t \in I What would the curve described by r(t) look like? The Attempt at a Solution...

You can give me a good examples where ##\frac{\partial}{\partial x}## is different to ##\frac{d}{dx}## ?

I am not quite sure how \frac{\partial}{\partial u}\left(\frac{\partial z}{\partial u}\right) =\frac{\partial}{\partial u}\left( u \frac{\partial z}{\partial x}+v\frac{\partial z}{\partial y} \right) comes to \frac{\partial z}{\partial x} + u\frac{\partial}{\partial u}\left(\frac{\partial...

Homework Statement y=1 / (cos h x), find dy/dx Homework Equations chain rule and coshx=(e^x+e^-x)/2 The Attempt at a Solution

is necessary to simplified the equation before differentiation? could I use the Quotient Rule without of simplifying ?

Problem statement Find dy/dx Revelant equations None Attempt at a solution This is what I got to so far but now I'm stuck... Any hints?

I have read about this method , and how feynman utilized this method. I like doing integrals for fun, but I can't seem to understand the conceptual idea on how to introduce a parameter into the integral. Can someone , in detail, explain to me how to introduce the parameter into the integral ...

Hi, I have x =(x^2+y^2)^[1/2] I differentiate 1= 1/2 (x^2+y^2)[-1/2] (2x+2yy') So far so good. I try to multiply this out. 1= (2x)/2 (x^2+y^2)[-1/2] + (2yy'/2)(x^2+y^2)[-1/2] I solve for y' y'= 1/{(x (x^2+y^2)[-1/2]} / {y(x^2+y^2)[-1/2] } 1/x (x^2+y^2)[1/2] * 1/y (x^2+y^2)[1/2] The...

Say we want to differentiate \arcsin x. To do this we put y=\arcsin x. Then x=\sin y \implies \frac{dx}{dy}= \cos y. Then we use the relation \sin^2 y + \cos^2 y = 1 \implies \cos y = \sqrt{1 - \sin^2 y} = \sqrt{1 - x^2}. Therefore \frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}}. My question is that...

Homework Statement If ##xs^2 + yt^2 = 1## (1) and ##x^2s + y^2t = xy - 4,## (2) find ##\frac{\partial x}{\partial s}, \frac{\partial x}{\partial t}, \frac{\partial y}{\partial s}, \frac{\partial y}{\partial t}## at ##(x,y,s,t) = (1,-3,2,-1)##. Homework Equations Pretty much those just listed...

I attached a picture of the problem from my online HW. I know how to solve the problem through direct differentiation, but that would too long to find the derivatives for this problem, and the problem actually suggests that I find another way. So my question is, what's the best way to solve this?

Homework Statement d/dθ csc-1(1/2)^θ = ? Homework Equations d/dx csc-1(x) The Attempt at a Solution I don't know how to deal with the exponent θ

Hello MHB members and friends!(Callme) An economy student asked me, if I could explain the following partial differentiation: \[\frac{\partial}{\partial C(i)}\int_{i\in[0;1]}[C(i)]^\frac{\eta - 1}{\eta}di =\int_{j\in[0;1]}[C(j)]^\frac{\eta - 1}{\eta}dj\frac{\eta -...

Here are the questions: I have posted a link there to this thread so the OP can see my work.

Homework Statement Show that a relation of the kind ƒ(x,y,z) = 0 then implies the relation (∂x/∂y)_z (∂y/∂z)_x (∂z/∂x)_y = -1 Homework Equations f(x,y) df = (∂f/∂x)_y dx + (∂f/∂y)_x dy The Attempt at a Solution I expressed x = x(y,z) and y = y(x,z) then found dx and...

Homework Statement z = x^2 +y^2 x = rcosθ y = rsinθ find partial z over partial x at constant theta Homework Equations z = x^2 +y^2 x = rcosθ y = rsinθ The Attempt at a Solution z = 1 + r^2(sinθ)^2 dz/dx = dz/dr . dr/dx = 2(sinθ)^2r/cosθ = 2tanθ^2x...

How do I compute the following differentiation by chain rule? \frac{d}{d\lambda}(\lambda^{-1}\phi(\lambda^{-1}x)) It is not a homework, but I can't figure out the exact way of getting the answer -\phi(x)-x^{s}\partial_{s}\phi(x)

I have a specific, for-learning-sake-only question on how the author of this link: http://www.math.ucla.edu/~yanovsky/Teaching/Math151A/hw5/Hw5_solutions.pdf gets past the details of the Intermediate Value Theorem on the following paragraph. If someone could fill in the details for me, it...

Homework Statement let V=f(x²+y²) , show that x(∂V/∂y) - y(∂V/∂x) = 0 Homework Equations The Attempt at a Solution V=f(x²+y²) ; V=f(x)² + f(y)² ∂V/∂x = 2[f(x)]f'(x) + [0] ∂V/∂y = 2[f(y)]f'(y) I'm sure I've gone wrong somewhere, I have never seen functions like this...

Is it possible to take momentum operator as dr/dt (r is position operator)? If not, why?

1) If u(r,\theta,\phi)=\frac{1}{r}, is \frac{\partial{u}}{\partial {\theta}}=\frac{\partial{u}}{\partial {\phi}}=0 because u is independent of \theta and \;\phi? 2) If u(r,\theta,\phi)=\frac{1}{r}, is: \nabla^2u(r,\theta,\phi)=\frac{\partial^2{u}}{\partial...

Helow! For a long time I aks me if exist differentiation/integration with respect to vector and I think that today I discovered the answer! Given: f(\vec{r}(t)) So, df/dt is: \bigtriangledown f\cdot D\vec{r} But, df/dt is: \frac{df}{d\vec{r}}\cdot \frac{d\vec{r}}{dt} This means that...

Homework Statement Homework Equations \frac{d}{dx} The Attempt at a Solution i couldn't do it because we didnt learn this type of question

I'm working my way through a solution of a problem and am confused on a step where a differentiation is performed. I'm sure I'm just forgetting some kind of rule, but I've been perusing my textbook and can't seem to figure out what I'm missing. Here's the step I'm talking about: Note that...

I got x = (u2 - v2) / u y = (v2 - u2) / v I differentiated them w.r.t u & v respectively & solved the given equation but I'm not getting the answer which is 0. Please view attachment for question!

Homework Statement Given that the surface x^{6}y^{5}+y^{4}z^{5}+z^{9}x^{7}+4xyz=7 has the equation z = f(x, y) in a neighborhood of the point (1, 1, 1) with f(x,y) differentiable, find: \displaystyle\frac{\partial^{2} f}{\partial x^{2}}(1,1) = ? Homework Equations The Attempt at a Solution...

Here are the questions: I have posted a link there to this topic so the OP can see my work.

Given a function: z(x,y) = 2x +2y^2 Determine ∂x/∂y [the partial differentiation of x with respect to y], Method 1: x = (z/2) - y^2 ∂x/∂y = -2y Method 2: ∂z/∂x = 2 ∂z/∂y = 4y ∂x/∂y = ∂x/∂z X ∂z/∂y = (1/2) X 4y = 2y One or both of these is wrong. Can someone point out...

Here are the questions: I have posted a link there to this topic so the OP can see my work.

Homework Statement Hello, I missed the class where we were introduced to implicit differentiation so have been catching up this evening. I think I have it, but please could you check my working? Thanks! Find the derivative of y2 = 2x + 1 \frac{d}{dx}([f(x)]^{2}) = \frac{d}{dx}([2x])...

Homework Statement The spherical head of a snowperson is melting under the HOT sun at the rate of -160 cc/h (cubic centimetres per hour.) Find the rate at which the radius is changing when the radius r=16. Use cm/h for the units. (The volume of a sphere is given by V= 4π⋅r^3/3.) I have...

Thank you for viewing my thread. I have been given the following steps for logarithmic differentiation: 1. Take natural logarithms of both sides of an equation y = f(x) and use the Laws of Logarithms to simplify. 2. Differentiate implicitly with respect to x. 3. Solve the resulting equation for...

find dy/dx: exy+x2+y2= 5 at point (2,0) I'm confused with finding the derivative with respect to x of exy. this is what I did so far for just this part: exy*d(xy)/dx exy*(y+x*dy/dx) do I need to put the parentheses on here? I thought so because that is the part where I used the product rule...

I just want to verify For Polar coordinates, ##r^2=x^2+y^2## and ##x=r\cos \theta##, ##y=r\sin\theta## ##x(r,\theta)## and## y(r,\theta)## are not independent to each other like in rectangular. In rectangular coordinates, ##\frac{\partial y}{\partial x}=\frac{dy}{dx}=0## But in Polar...

I have another two problems I find difficult. They both involve trigonometry, so I thought I could fit both under the same post. Also, if possible, I'd like some help in regards to confirming that one problem I've solved is done correctly. Homework Statement First, the derivative. Find y'...

##r^2=x^2+y^2\;\Rightarrow \; 2r\frac{dr}{dx}=2x\;\Rightarrow\; \frac{dr}{dx}=\frac{x}{r}## Then is it true ##\frac{dx}{dr}=\frac{r}{x}##? I am not sure this is correct as r^2=x^2+y^2\;\Rightarrow \; 2r=2x\frac{dx}{dr}+2y\frac{dy}{dr}

Homework Statement Differentiate the following with respect to x y = \frac{4}{x^{3}} + \frac{x^{3}}{4}The Attempt at a Solution So the problem here is really getting this into a form that is easy to differentiate and i'd just like to show what I'm doing before I go ahead and do the rest of...