Derivative Definition and 1000 Threads

Homework Statement I know this is more of a physics question, but I tried there and wasn't successful. I have done a physics experiment measuring the weight as a function time of the discharge of water from a cylindrical bottle with a pinhole at the bottom. What I ultimately want to get at is...

I'm trying to find the partial derivatives of: f(x,y) = ∫ (from -4 to x^3y^2) of cos(cos(t))dt and I am completely lost, any help would be appreciated, thanks.

C \in \mathbb{R}^{m \times n}, X \in \mathbb{R}^{m \times n}, W \in \mathbb{R}^{m \times k}, H \in \mathbb{R}^{n \times k}, S \in \mathbb{R}^{m \times m}, P \in \mathbb{R}^{n \times n} ##{S}## and ##{P}## are similarity matrices (symmetric). ##\lambda##, ##\alpha## and ##\beta## are...

Homework Statement Find the directional derivative of ##f## at ##P## in the direction of ##a##. ## f(x,y) = 2x^3y^3 ; P(3,4) ; a = 3i - 4j ## Homework Equations ## D_u f(x_0, y_0, z_0) = f_x(x_0, y_0, z_0)u_1 + f_y(x_0, y_0, z_0)u_2 ## The Attempt at a Solution ## f_x (x,y) = 6x^2y^3##...

Let's say ##f(x)=ax^2##. Then ##d^2f/dx^2=2a##. Now we can make the change of variables ##y\equiv\sqrt ax## to give ##f(y)=y^2##. Then ##d^2f/dy^2=2##. It follows that ##\frac{d^2f}{dx^2}=a\frac{d^2f}{dy^2},## but I can't replicate this with the chain rule. I would put...

Homework Statement Find the derivative of the function y = (3-2x^3+x^6 )/x^9 Homework Equations Derivatives The Attempt at a Solution I have tried to use the quotient rule and got to -6x^11 + 6x^14 - 27x^8 + 18x ^24 - 9x ^14 / (x^9)^2 Which doesn't look close to the answer -27/x^10 +...

Homework Statement At time t, the position of a body moving along the s-axis is s= t^3 -12t^2 + 36t m(meters) Find the total distance traveled by the body from t = 0 to t = 3. Homework Equations Derivatives The Attempt at a Solution I got the derivative which is 3t^2 - 24t + 36(meters) I...

Sorry about the title, had a hard time trying to fit the question on the given space. The question is quite simple : If F = F(x_1,...,x_n,t) , Under what conditions is \frac{d }{dt} \frac{\partial F }{\partial xi} = \frac{\partial }{\partial xi} \frac{dF }{dt} true?

Hello, I've been studying PID control and I've undestrood many things, but in every source I've read there is no exact definition for what the Integral Time and Derivative Time are. I now know what is the results of setting them high and low—to some extent—and have studied a bit the tuning...

Homework Statement The problem is hopefully attached, I had to take a screen shot. Homework Equations I understand the process of taking the derivative of position to get velocity. *refer to derivative rules... for example r(x)=2x^2-6x+8 therefore r`(x)=4x-6 The Attempt at a Solution I am...

Hi, Is there a theorem that says that if f(n) = g(n) and f'(x) >= g'(x) for each x > n, then it means that for each x>n f(x) >= g(x)? or is there a theorem that required more properties of g and f that implies so? Thanks!

Homework Statement 1/2mz^2 +mgh=mgh-zero , get g The Attempt at a Solution z= velocity z^2=g(2h0-2h) if i set z^2=a 2h0=b (nonvariable) 2h=c a=g(b-c) y'=-g Can i then say that dz^2/d2h = -g I wonder if every step is correct, The writing inbetween is not very important! I mostly...

(Hope it's okay that I'm posting so much at the moment, I'm having quite a bit of trouble with something I'm doing) Homework Statement I'm having trouble with the simplification of the following equation. The answer is shown, but I can't figure out the process to get to it. \frac{d}{dt}...

Is Impulse an anti derivative of momentum? I know that momentum is an anti derivative of force (proof below), but I'm struggling with understanding the difference between momentum and impulse. My thoughts led me to think that both impulse and momentum are anti derivatives of force, but I'm not...

Homework Statement Find the first and second derivative of the following function: F(x)=e4ex Homework Equations d/dx ex = ex d/dx ax = axln(a) The Attempt at a Solution I know the derivative of ex is just ex, but I'm not sure how to go about starting this one. I'm near certain I need to use...

Quick question (a little rusty on this): Why don't unit vectors in Cartesian Coordinates not change with time? For example, suppose \mathbf{r} (t) = x(t) \mathbf{x} + y(t) \mathbf{y} + z(t) \mathbf{z} How exactly do we know that the unit vectors don't change with time? Or in other words...

Hey Guys! I was working on an integration problem, and I "simplified" the integral to the following: $$\int \limits_0^{2\pi} (7.625+.275 \cos(4x))^{1.5} \cdot (A \cos(Nx) + B \sin(Nx)) \cdot (Z-v \cos(x)) dx$$ This integral may seem impossible (I have almost lost all hope on doing this...

Hello, I have this problem \frac{\partial}{\partial\,x}\int_0^{∞}\log(1+x)\,f_X(x)\,dx, where x is a random variable, and f_X(x) is its probability density function. It's been a long time since I encountered a similar problem, and I forgot how to do this. Do we use Leibniz integral rule...

Just using basic dimensional analysis, it appears the time derivative of centripetal acceleration is ## \vec{r} \omega^3 ##, but this intuitive guess would also extend to higher order time derivatives, no? Implying: ## \frac {d^n \vec{r}}{dt^n} = \vec{r} \omega^n ## It seems to follow from the...

In trying to get an intuition for curl and divergence, I've understood that in the case of R2, div f(x,y) = 2Re( d/dz f(z,z_)) and curl f(x,y) = 2Im( d/dz f(z,z_)), where f(z,z_) is just f(x,y) expressed in z and z conjugate (z_). Is there any way of proving the fundamental properties of div and...

Since lnx is defined for positive x only shouldn't the derivative of lnx be 1/x, where x is positive. My books does not specify that x must be positive, so is lnx differentiable for all x?

I have been playing around with calculus for a while and I wondered what would it be like to make some changes to the definition of derivatives. I'd like to look at the original definition of derivatives in this way (everything is in lim Δx→0): F(x+Δx) - F(x) = F'(x) * Δx The Δx factor...

Given $ f(z) = e^{-\frac{1}{z}} $, find f'(z) and identify the maximal region within which f(z) is analytic I found f'(z) = \frac{e^{-\frac{1}{z}}}{z^2} , is that right? I think I should be using the Cauchy-Riemann Conditions to check if analytic, but this function is not in the form u+iv...

Homework Statement Basically, I'm looking at the property that says if the magnitude of a vector valued function is constant, then the vector function dotted with it's derivative will be zero. But I'm stuck towards the end because the proof I found online seems to skip a step that I'm not...

Call me crazy, but I do recall the power rule of integration viz: f(x) = x^n, f(x)' = n*x^n-1. Therefore, it seems as though 2x^2 would have a derivative of 4x. Fine. So why have I encountered someone else claiming that it's 8x? WHAT?! Who's right?

A lot of web pages/books show how to use cosx=sin(Pi/2-x) and the chain rule to prove that the derivative of cosx=-sinx. My question is how to use this identity and the defintion of the derivative to prove the same thing. Or whether it is at all possible. Seeing that i get...

In the book that I am reading the derivative of y = bpx is bpxxlogep. How?

If I have an equation where there is a derivative surrounding the variable, how do I undo the derivative and solve for the variable? Example would be- A= dx/dy when x=m*v*λ-2 and y=y Solving for v. I am a beginner so please explain thoroughly.

I can't figure out why my demonstration of snell's law fails, that's the demonstration: (I used a photo) I think it fails because the function t (HO) represents a line and so the concept of minimum is not defined, when I take the derivative and equal it to 0 I'm considering the case when the...

Hi guys, I'm working with this interaction Lagrangian density ##\mathcal{L}_{int} = \mathcal{L}_{int}^{(1)} + \mathcal{L}_{int}^{(2)} + {\mathcal{L}_{int}^{(2)}}^\dagger = ia\bar{\Psi}\gamma^\mu\Psi Z_\mu +ib(\phi^\dagger\partial_\mu \phi - \partial_\mu\phi^\dagger \phi)Z^\mu,## with ##...

Homework Statement the original function is ##−6 x^3−3x−2 cosx## ##f′(x)=−2x^2−3+2sin(x)## ##−2x^2 ≤ 0## for all x and ##−3+2 sin(x) ≤ −3+2 = −1##, for all x ⇒ f′(x) ≤ −1 < 0 for all x The Attempt at a Solution this problem is part of a larger problem which says there is a cubic...

Homework Statement f(x)= (sin x)^(sin x) Homework EquationsThe Attempt at a Solution Taking logarithm on both sides I get: ln y = ln ((sin x)^(sin x))

Homework Statement Given n=(x + iy)/2½L and n*=(x - iy)/2½L Show that ∂/∂n = L(∂/∂x - i ∂/∂y)/2½ and ∂/∂n = L(∂/∂x + i ∂/∂y)/2½ Homework Equations ∂n Ξ ∂/∂n, ∂x Ξ ∂/∂x, as well as y. The Attempt at a Solution ∂n=(∂x + i ∂y)/2½L Apply complex conjugate on right side, ∂n=[(∂x + i ∂y)/2½L] *...

Homework Statement From the transformation from polar to Cartesian coordinates, show that \begin{equation} \frac{\partial}{\partial x} = \cosφ \frac{\partial}{\partial r} - \frac{\sinφ}{r} \frac{\partial}{\partialφ} \end{equation} Homework Equations The transformation from polar to Cartesian...

I think it is not true that a discontinuous ##\nabla^2\psi## implies a discontinuous ##\nabla\psi##, because a continuous function can have a discontinuous derivative, eg. ##y=|x|##. Is it true that ##\nabla\psi## must always be undetermined at the boundary where ##V=\infty##? Attached below...

Find the function with the given derivative whose graph passes through point P. $$r'\left(\theta\right) =6+\sec^2 \left({\theta}\right), P\left(\frac{\pi}{4},0\right)$$ 6+sec^2(x) The phase shift appears to be 1 but not sure how to get that How do add another equation to desmos?

hello! 1) what is the process to get the derivative of an equation that requires you to do first the chain rule and then the product/quotient rule, eg. sin(x^2(x+1))? 2) what is the process to get the derivative of an equation that requires you to do first the product/quotient rule and then the...

Hello! I'm trying to find the 2nd derivative of y(t)=tan5t. I first found the first derivative.. and got y'(t)=sec^2(5t)(5) --> 5sec^2(5t) --> 5/(cos^2(5t) But to find the 2nd derivative I'm confused... I got until y"(t)=\frac{cos^2(5t)(5)'-(5)(cos^2(5t))'}{(cos^2(5t)(cos^2(5t))}

x2y2 + (y+1)e-x=2 + x Defines y as a differentiable function of x at point (x, y) = (0,1) Find y′: My attempt: ∂y/∂x =2xy3 + (-y-1)e-x=1 ∂y/∂y = 3x2y2 - e-x=0 Plugging in for x and y ⇒ ∂y/∂x = -3 ∂y/∂x = -1 For some reason I think y′ is defined as (∂y/∂x) /(∂y/∂y) = 3 At leas this give...

Hello. I have a question regarding curvature and second derivatives. I have always been confused regarding what is concave/convex and what corresponds to negative/positive curvature, negative/positive second derivative. If we consider the profile shown in the following picture...

I am currently studying optical microscope and discover that the axial resolution is limited as r(z) = 2pi / (NA)^2. However, while I got hints that it is due to the Rayleigh's limit, I can't derivative the equation using numerical method. It would be huge thanks if anyone can help me on the...

i have a mathematical question which is quite similar to one asked before, still a bit different https://www.physicsforums.com/threads/derivative-of-first-term-in-lagrangian-density-for-real-k-g-theory.781472/the first term of KG-Lagrangian is: \frac{1}{2}(\partial^{\mu} \phi)(\partial_{\mu}...

Homework Statement Find all stationary points of the function G(x, y) = (x^3)*e^(−x^2−y^2) Homework Equations fx=0 and fy=0 The Attempt at a Solution Gx = 3x^2*e^(-x^2-y^2) +x^3(-2x)e^(-x^2-y^2) = e^(-x^2-y^2)(3x^2-2x^4) Gx = 0 implies 3x^2-2x^4=0 x^2(3-2x^2)=0 hence x =0 ,+or-...

Homework Statement f(c,l) = log(c - ψ(1-l)^θ ) What is the derivative of this function wrt. l and c?Homework Equations I know that the derivative of log (x) = 1/x The Attempt at a Solution I got wrt c: 1/ c - ψ(1-l)θ and wrt l: θψ(1-l)^θ-1 / c - ψ(1-l)^θ

Hey I'm trying to follow the derivation given here: http://lampx.tugraz.at/~hadley/ss1/studentpresentations/Bloch08.pdf Homework Statement As it says in the pdf: "Based on Noether's theorem construct the energy-momentum tensor for classical electromagnetism from the above Lagrangian. L=-1/4...

Having a melt down as I have done this problem twice now and my exam is tomorrow and I can't seem to figure it out anymore... ugh. 1. Homework Statement The depth of a lake at the point on the surface with coordinates (x, y ) is given by D(x, y ) = 100−4x 2 −y 2 . a) If a boat at the point (−1...

So I know that momentum is the time derivative of force, but what is the time derivative of force? That is, p=mv, f=ma, ?=mj (if j is jerk/jolt). Thanks!

So looking through my notes I can't seem to understand how to get from one step to the next. I have attached a screenshot of the 2 lines I'm very confused about. Thanks. BTW: The equations are for the log likelihood in a mixture of gaussians model EDIT: To elaborate I am particularly...

I'm relatively new to calculus and I have a new chapter in my study which is on the Implicit Function, Implicit Differentiation and Higher Derivatives of a function, the problem is I don't understand the meaning of a 2nd or 3rd or whatever the higher derivative of a function is, what I know is...

Homework Statement Show that δ(x-x') = d/dx Θ(x-x') Homework Equations ∫ f(x') δ(x-x') dx' = f(x) Θ(x-x') vanishes if x-x' is negative and 1 if x-x' is positive The Attempt at a Solution I saw a relation of the δ function but I don't know why is it like that. Integral of δ(x-x') from -∞ to x...