Derivatives Definition and 1000 Threads

I'm thinking in particular about Lenny Susskind's lectures, but I've seen other lecturers do it too. They'll be writing equation after equation using the partial derivative symbol: \frac{\partial f}{\partial a} And then at some point they'll realize that some problem they're currently...

Hello, Given is the function f = f(a,b,t), where a=a(b) and b = b(t). Need to express first and second order derivatives. \frac{\partial f}{\partial a} and \frac{\partial f}{\partial b} should be just that, nothing more to it here, correct? But \frac{df}{dt} = \frac{\partial...

Homework Statement Differentiate f(x) = x^{1/2} - x^{1/3} Homework Equations f(x) = f'(x)- g'(x) The Attempt at a Solution I am a little stuck about what to do after the first couple steps. Here is my attempt. f(x) = x^{1/2} - x^{1/3} f'(x) = (x^{1/2})' -(x^{1/3})' =...

Homework Statement Differentiate f(x) = (x^2 - 3x)^2 Homework Equations f'(x) = nf'(x)f(x)^(n-1) The Attempt at a Solution f’(x) = 2(x2-3x)’(x2-3x)2-1 = 2(2x-3)(x2-3x)1 = 2(2x3 – 6x2 – 3x2 + 9x) = 2(2x3 – 9x2 + 9x) = 4x3 – 18x2+ 18x Is this correct?

Homework Statement Expand x/(x-1) at a=1 The book already gives the expansion but it doesn't explain the process. The expansion it gives is: x/(x-1) = (1+x-1)/(x-1) = (x-1)^(-1) + 1 Homework Equations The Attempt at a Solution I've already solved for the Mclaurin expansion for the same...

I've had to hit my books to help someone else. Ugh. Say we have the coordinate transformation \bf{x}' = \bf{x} + \epsilon \bf{q}, where \epsilon is constant. (And small if you like.) Then obviously d \bf{x}' = d \bf{x} + \epsilon d \bf{q}. How do we find \frac{d}{d \bf{x}'}? I'm missing...

Homework Statement Let f(x) = 2x2 -3x -5. Show that the slope of the secant line through (2, f(2)) and (2+h, f(2+h)) is 2h + 5. Then use this formula to compute the slope of : (a) The secant line through (2, f(2)), and (3, f(3)) (b) The tangent line at x = 2 (by taking a limit)...

Homework Statement The separation of layers is considered to occur at the thermocline, which is defined as the location of the steepest slope in the temperature gradient. Mathematically, this occurs at the inflection point – so the position of the thermocline can be found from the following...

For some polynomial functions it is useful to logarithmize both sides of the eq. First. How can this be applied for inverse trig functions? Is it even possible?

1. Is this the only example of a function ##f(x) \in C^1([0,1])## with discontinuous derivative $$f(x) = \begin{cases} x^2 sin(\frac{1}{x}) & \textrm{ if }x ≠ 0 \\ 0 & \textrm{ if }x = 0 \\ \end{cases}$$ It seems this example is over-used. Do we have other examples besides this one in...

Homework Statement If z=f(x,y) with u= x^2 -y^2 and v=xy , find the expression for (∂x/∂u). the (∂x/∂u) will be used to calsulate ∂z/∂u. my question is how to find (∂x/∂u). I don't know what to keep constant. Maybe the question has some problem. The answer is (∂x/∂u)=(x/2)/(x^2+y^2)...

Homework Statement We want to make a conical drinking cup out of paper. It should hold exactly 100 cubic inches of water. Find the dimensions of a cup of this type that minimizes the surface area. Homework Equations SA = pi*r^2 + pi*r*l where l is the slant height of the cone. V =...

Hi, I've recently taken a Calculus 1 (Differential Calculus) course and I've been looking ahead to see what sort of material is covered in the Calculus 2 (Integral Calculus) course. I am wondering about the relationship between derivatives and integrals. From what I understand, an integral...

Hello guys! Lately I've been studying some topics in Physics which require an extensive use vector calculus identities and, therefore, the manipulation of partial redivatives of vectors - in particular of the position and velocity vectors. However, I am not sure if my understanding of partial...

Homework Statement Where T(x,t)=T_{0}+T_{1}e^{-\lambda x}\sin(\omega t-\lambda x) \omega = \frac{\Pi}{365} and \lambda is a positive constant. Show that T satisfies T_{t}=kT_{xx} and determine \lambda in terms of \omega and k. I'm not to sure what is meant by the latter part of "determine...

f(z,t)=\frac{A}{b(z-vt)^{2}+1}... \frac{\partial^{2} f(z,t)v^{2} }{\partial z^2}=\frac{-2Abv^{2}}{[b(z-vt)^{2}+1]^{2}}+\frac{8Ab^{2}v^{2}(z-vt)^{2}}{[b(z-vt)+1]^{3}}=\frac{\partial^2 f}{\partial t^2} \frac{-2Abv^{2}}{[b(z-vt)^{2}+1]^{2}}+\frac{8Ab^{2}v^{2}(z-vt)^{2}}{[b(z-vt)+1]^{3}} this...

Homework Statement An object is traveling along a linear path according to the equation s(t) = 4t^3 - 3t^2 + 5 where t is measured in seconds and s(t) measured in meters. How far has the object traveled when its acceleration is zero? Homework Equations The Attempt at a Solution...

Newton's laws says ## F=ma ##. Which, as far as I can see, states that all physical interactions concern the second time derivative of position. And because there is no other way for two bodies to interact in the physical world, the "worst" I can do to a system is change its acceleration, right...

I have z=(e^y)φ*[y*e^(x^2/2y^2)].I have to prove that y*(dz/dx) -x*(dz/dy)=0.First of all what does φ mean there?

Hi, here I come again now with a problem, this time is more because the boo didn't really explain much deeper than the minimum. This time it's not homework related ... Yet. Lol. I just wanted to know how I would separate an equation like y=sin^2x cos3x To find the derivativ; I guess how...

Homework Statement Homework Equations I can't think of much. u(x,y) is harmonic, so its double derivatives with respect to x and y add up to zero. I'm not 100% sure, but does being harmonic also imply that u satisfies the cauchy riemann equations? That might come in handy in the...

Here's an example from my homework. I already turned it in, though. I basically just copied what I could from my notes, but I have no idea how this is done. Could someone explain this to me? I can't find anything intelligible (at least to me) of this stuff on any website. My notes contain parts...

Hello PH, This is my first post. I came here while studying partial derivatives and after clicking here and there for over 4hrs for an answer. While practicing the derivatives rules i came across the hideous quotient rule. I've solved around 20 fractional problems trying to find a decision...

The idea of varying one thing but keeping others constant is central in scientific analysis. People want to know, other things constant, the effect of taking vitamins, smoking or drinking alcohol, just as examples. Is the idea of the partial derivative analogous to scientific empiricism's...

We know that the first derivative represents the slope of the tangent line to a curve at any particular point. We know that the second derivative represents the concavity of the curve. Or, the first derivative represents the rate of change of a function, and the second derivative represents the...

Derivatives with Quotient Law Help! I have a test tomorow, any help is much appreciated! :) Homework Statement Dervive using the quotient rule: [(2-x)^3] / [(x+1)^2] My attempt: = [(x+1)^2 (3(2-x)^2)]-[(2-x)^3 2(x+1)] When I try expanding I get the wrong answer. The...

Homework Statement Which of the following satisfy (f^k)(x) = 0 for all k >= 6? a) f(x) = 7x^4 + 4 + x^-1 b) f(x) = sqrt(x) c) f(x) = x^(9/5) d) f(x) = x^3 - 2 e) f(x) = 1 - x^6 f) f(x) = 2x^2 + 3x^5 Homework Equations None, but given as a problem in a chapter where the topic is higher...

Hello, Today did me and my friend talked of derivate and he asked about some help. Then he asked me is it possible to derivate $y^3+5x^2=5x-2y$ and i was clueless how i derivate that. Is this difficoult to derivate?is it possible to do it?

Homework Statement This is not really a homework or a coursework question. But I got a warning that I should submit my post in this section of the website.. I'm just saying this because I don't know if the answer to my question is at all achievable. And if it is how I should go about trying to...

I have given a function g(t)=∇(f(x(t))) , f: IR³->IR and x: IR-> IR³ and want to express the first 3 derivatives with respect to time most simply. I thought that g'(t)=Hessian(f(x(t)))dx/dt but how do I get the further derivatives. is there any chance to express those in terms of taking the...

Here's a question. This formula seems to be the keystone of calculus. That seems to be what the calculus books say, and it makes sense to me, as a rank beginner. This equation is what makes the seeming magic of defining the slope of a dimensionless point on a curved slope possible. And...

Homework Statement Find the coordinates of the point(s) on the following curves where the second derivative is as stated. Homework Equations y= \frac{x^3}{12} and \frac{d^{2}y}{dx^{2}} = 1.5 The Attempt at a Solution I'm used to working with the first derivative. Would I need to...

Homework Statement general course question Homework Equations N/A The Attempt at a Solution fx is a first order partial derivative fxy is a second order partial derivative fxyz is a third order partial derivative I understand that Clairaut's Theorem applies to second order...

f(x)=5x^3+6x^2-3x+lnx (lnx)`=1/x f(x)=2x^4+3x^2+ cosx (cosx)`=-sinxI know that if I only have x, like 3x, then x disappears (correct me if I'm wrong). So what happens with lnx if x disappears? Same thing with cosx. The lesson is extreme values of functions and i saw critical points mentioned...

I am utilitizing rotation vectors (or SORA rotations if you care to call them that) as a means of splitting 3D rotations into three scalar orthogonal variables which are impervious to gimbal lock. (see SO(3)) These variables are exposed to a least-squares optimization algorithm which...

In a thermodynamics question, I was recently perplexed slightly by some partial derivative questions, both on notation and on physical meaning. I believe my questions are best posed as examples. Suppose we have an equation, (\frac{\partial x(t)}{\partial t}) = \frac{1}{y}, where y is a...

Homework Statement compute the following derivatives using the product rule and quotient rule as necessary, without using chain rule. Homework Equations d/dx ((sin(x))^2) The Attempt at a Solution =(sin(x))(sin(x)) =(cos(x))(sin(x))+(sin(x))(cos(x)) =2(sin(x))(cos(x))

Why do some functions not have Anti derivatives?? as the title says why are some functions like ## √cotx##(root cotx) not integratable??

Hi i have two questions: 1) When asked to prove \mathcal{L}_{u}\mathcal{L}_{v}W - \mathcal{L}_{v}\mathcal{L}_{u}W = \mathcal{L}_{[u,v]}. I achieved [u,v]w = \mathcal{L}_{[u,v]}. This was found by appliying a scalar field <b> to the LHS and simplifying and expanding using + and scalar...

Hi everyone, Molecules move into a vacuum chamber from an oven at constant T. The molecules then pass through a slit. They reach two rotating discs before finally reaching a detector. Show that a molecule that passes through the first slit will...

Hi, I am trying to find stationary points of the function f(x)=(xtAx)/(xtx) (the division of x transpose times A times x divided by x transpose x) where A is a px1 symmetric matrix. I need to take the derivative of this to show that when i set it to zero i get the eigenvectors of A. I know how...

Hi, I didn't put this into homework since is only a question about notation: In a problem, given a Lagrangian and a transformation (x,y) -> (x',y'), where these x' and y' depend on λ, in particular like e^{\lambda}. The problem asks for the derivative \frac{\delta L}{\delta \lambda}. What...

Homework Statement Of all cylinders inscribed in sphere of radius R largest area of side(M) has cylinder which hight is R\sqrt{2}. Prove. The Attempt at a Solution I understand how to prove this i only have problem with derivative: M=2*r*Pi*H and r=\frac{\sqrt{(2R)^2 - H^2}}{2}...

Anyone how to evaluate this integral? \int_{-1}^{1} (1-x^2) P_{n}^{'} P_m^{'} dx , where the primes represent derivative with respect to x ? I tried using different recurrence relations for derivatives of the Legendre polynomial, but it didn't get me anywhere...

let f = x2 + 2y2 and x = rcos(\theta), y = rsin(\theta) . i have \frac{\partial f}{\partial y} (while holding x constant) = 4y . and \frac{\partial f}{\partial y} (while holding r constant) = 2y . i found these partial derivatives by expressing f in terms of only x and y, and then in...

Homework Statement Being a>0 and f:[a,b]--->R continuos and differentiable in (a,b), show that there exists a t ##\in## (a,b) such that: ## \frac{bf(a)-af(b)}{b-a}=f(t)-tf'(f)## The Attempt at a Solution For lagrange's theorem, we have: ## \frac{f(a)-f(b)}{b-a}= -f'(t) ## thought i could...

Hi everyone, I am working on simplifying a differential equation, and I am trying to figure out if a simplification is valid. Specifically, I'm trying to determine if: \frac{\del^2 p(x)}{\del p(x) \del x} = \frac{\del^2 p(x)}{\del x \del p(x)} where p(x) is a function of x. Both p(x)...

Homework Statement Hey, can I just check these functional derivatives?: 1) \frac{\delta F[g]}{\delta g(y)} where F[g] = \int dx \left[ \frac{1}{\sqrt{1+(g'(x))^2}} - 2g(x) + 5 \right]\>. 2) \frac{\delta F[a,b,g]}{\delta g(y)} where F[a,b,g] = \int d^4x \left[ A(\partial_{\mu}...

Homework Statement Determine whether a function with partial derivatives f_x(x,y)=x+4y and f_y(x+y)=3x-y exist. The Attempt at a Solution The method I've seen is to integrate f_x with respect to x, differentiate with respect to y, set it equal to the given f_y and show that it can't be...

I'm working on a calculus project and I can't seem to work through this next part... I need to substitute equation (2) into equation (1): (1): r\frac{\partial}{\partial r}(r\frac{\partial T}{\partial r})+\frac{\partial ^{2}T}{\partial\Theta^{2}}=0 (2): \frac{T-T_{0}}{T_{0}}=A_{0}+\sum from n=1...