Derivatives Definition and 1000 Threads

Homework Statement Newton devised the following method for approximating a real root of the equation f(x) = 0. i.e. a real number for which f(r) = 0. We begin by guessing an approximation, say x1, to the real root r. (i) Find the equation of the line tangent to the graph of y = f(x) at the...

Hi! I think I've managed to solve this problem, but I'd like it to be checked Homework Statement show that if $$f : A\subset \mathbb{R}\to \mathbb{R}$$ and has both right derivative: $$f_{+}'(x_0),$$ and left derivative $$f_{-}'(x_0)$$ in $$x_0\in A$$, then $$f$$ is continuos in $$x_0.$$...

Suppose I have a transformation (x'_1,x'_2)=(f(x_1,x_2), g(x_1,x_2)) and I wish to find \partial x'_1\over \partial x'_2 how do I do it? If it is difficult to find a general expression for this, what if we suppose f,g are simply linear combinations of x_1,x_2 so something like ax_1+bx_2 where...

Could someone please explain how the formula at the bottom of the page is derived i.e. how is the Taylor theorem used to obtain it ?

Homework Statement Suppose f: R -> R is differentiable and let h(x,y) = f(√(x^2 + y^2)) for x ≠ 0. Letting r = √(x^2 + y^2), show that: x(dh/dx) + y(dh/dy) = rf'(r) Homework Equations The Attempt at a Solution I have begun by showing that rf'(r) = sqrt(x^2 + y^2) *...

Object rotation about a fixed axis?? question about derivatives in this problem?? An object rotates about a fixed axis, and the angular position of a reference line on the object is given by θ=0.40e^(2t), where θ is in radians and t is in seconds. Consider a point on the object that is 4.0 cm...

Homework Statement I have the professor's solutions for a homework we handed in. There is a part that is confusing me. We have the following equation: $$E = \frac{1}{2}(m_1 + m_2)\dot{x}^2-(m_1-m_2)gx$$ Homework Equations We want to find: $$dE/dt = 0$$ The Attempt at a Solution...

what does d/ds (e^s cos(t)du/dx + e^s sin(t)du/dy) give, given that u = f(x,y) i don't know how to manipulate d/ds and how to derive using d/ds. i am trying to simplify the expression, but i don't know, i just get stuck in the middle of can't get farther than here...

Homework Statement I'm taking a fluid mechanics class and I'm having an issue with acceleration and background knowledge. I know this is ridiculous, but I was hoping someone might be able to explain it for me. Homework Equations I definitely understand: ##a=\frac{d\vec{V}}{dt}## And I...

Homework Statement find d/dt for a rectangle.Homework Equations A=bh product rule for derivatives (the first times the derivative of the second plus the second times the derivative of the first) Chain rule for derivatives The Attempt at a Solution If b is a constant, then I know that dA/dh = b...

1. Homework Statement [/b] Let z=3x2-y2. Find all points at which magnitude(nabla Z)=6 First things first I took the partials dz/dx and dz/dy dz/dx=6x dx/dy=-2y I know that √(36x2+4y2)=6 or (36x2+4y2)=36 Then using the above relation I solved for each variable getting 1.y=√(9-x2)...

http://www.mathway.com/math_image.aspx?p=f%28x%29SMB01%28SMB02RSMB03xSMB02rSMB03+SMB02FSMB031SMB10SMB02RSMB03xSMB02rSMB03SMB02fSMB03%29SMB02ESMB033SMB02eSMB03?p=117?p=42, show that...

Find the two first-order partial derivatives of z with respect to x and y when z = z(x, y) is defined implicitly by z*(e^xy+y)+z^3=1. I started by multiplying the brackets out to give; ze^xy + zy + z^3 - 1 = 0 i then differentiated each side implicitly and got; dz/dx = yze^xy and...

Homework Statement If f has a finite third derivative f''' on (a,b) and if f(a)=f'(a)=f(b)=f'(b)=0 prove that f'''(c)=0 for some c in (a,b) Homework Equations Rolle's Theorem: Assume f has a derivative (finite or infinite) at each point of an open interval (a,b) and assume that f is...

Homework Statement let u be a function of x and y.using x=rcosθ y=rsinθ,transform the following expressions in the terms of partial derivatives with respect to polar coordinates:(d^u/dx^2(double derivative of u with respect to x)+d^2u/dy^2(double derivative of u with respect to y)...

Hi guys I have 5 term which each term includes three function of time (autonomous system ), i want to obtain time derivative of each term as other terms for example if i define each term as a dimensionless variable , e.g. x , y , z , v , w , i want to know dx/dt = as a combination of x , y ...

Homework Statement If z=\frac{1}{x}[f(x-y)+g(x+y)], prove that \frac{\partial }{\partial x}(x^2\frac{\partial z}{\partial x})=x^2\frac{\partial^2 z}{\partial y^2} Homework Equations The Attempt at a Solution I don't know how I'm supposed to find the partial derivative of z with respect to...

Homework Statement Determine the lowest derivative order for which the limit towards 0+ of the nth order derivative of f is nonzero (or otherwise does not exist). f = e^{\frac{-1}{x^{2}}} Homework Equations lim_{x\rightarrow0+}\frac{d^{n}}{dx^{n}}e^{\frac{-1}{x^{2}}} The Attempt at...

Homework Statement If f(x,y,z) = 0, then you can think of z as a function of x and y, or z(x,y). y can also be thought of as a function of x and z, or y(z,x) Therefore: dz= \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y} dy and dy= \frac{\partial y}{\partial x}dx +...

Homework Statement Taking k and ω to be constant, ∂z/∂θ and ∂z/∂ф in terms of x and t for the following function z = cos(kx-ωt), where θ=t2-x and ф = x2+t. Homework Equations The Attempt at a Solution I'm finding this difficult as t and x are not stated explicitly. I know how to...

Homework Statement Find all first and second partial derivatives of the following function: z = e^(-ET) where E and T are functions of z. I know how to do partial differentiation, but not when the variables are functions of z? I don't understand - is there some sort of implicit...

Homework Statement The surface z=f(x,y)=√(9-2x2-y2) and the plane y=1 intersect in a curve. Find parametric equations for the tangent line at (√(2),1,2).Homework Equations Partial derivativesThe Attempt at a Solution Okay, so I'm just trying to work through an example in my textbook, so...

1. If F(x) = f(xf(xf(x))), where f(1) = 2, f(2) = 3, f '(1) = 4, f '(2) = 5, and f '(3) = 6, find F'(1). I feel I have a decent grasp on the chain rule, product rule, etc, but when faced with a problem like this I just blank out. I don't even really know where to begin. Unfortunately I...

Homework Statement Let f(x,y,z)=u(t), where t=xyz. Show that f_{xyz} = F(t) and find F(t). The Attempt at a Solution I'm a little confused about the presentation of the variables in this problem. What does F(t) refer to? This isn't a chain rule question, because it's presented before chain...

Homework Statement f'_x = kx_k, k = 1, 2, ..., n The Attempt at a Solution The partial should be f(sub)x(sub)k, as in, the partial derivative of f with respect to x_k. I wasn't sure how to represent that using TeX. I'm honestly at a complete loss here, because I'm not entirely sure what the...

Derivatives of logarithmic functions - please help Homework Statement I am having trouble differentiating logarithmic functions. I am following this basic rule they gave us, namely: if y = ln g(x) then y' = g'(x)/g(x), but it is not working :(. Where am I going wrong? Homework Equations...

Homework Statement I have an expression for the partial derivative of u with respect to s, which is \frac{\partial\,u}{\partial\,s} = \frac{\partial\,u}{\partial\,x}x + \frac{\partial\,u}{\partial\,y}y How do I compute \frac{\partial^2u}{\partial\,s^2} from this?

I'm curious how the derivative of a complex function can be represented visually. It is defined as the limit of (f(z_{0} + Δz) - f(z_{0}) / Δz as Δz approaches 0. Is it right to say that f(z_{0} + Δz) represents a neighborhood of radius Δz around z_{0} in this case? Does the derivative still...

Homework Statement F(3)=−2, g(3)=9, f′(3)=−2, and g′(3)=2, find the following numbers: (a) (f+g)′(3) (b) (fg)′(3) (c) (f/g)′(3) (d) (f/(f−g))′(3)The Attempt at a Solution I already have (a) and (b) [a=0 and b=-22] for (c) i tried: (g(x)*f'(x) - f(x)g'(x)) / (g(x))^2 evaluate at 3...

$x = r\cos\theta$ and $y=r\sin\theta$ $$ \frac{\partial u}{\partial\theta} = \frac{\partial u}{\partial x}\frac{\partial x}{\partial\theta} + \frac{\partial u}{\partial y}\frac{\partial y}{\partial\theta} = -r\sin\theta\frac{\partial u}{\partial x} + r\cos\theta\frac{\partial u}{\partial y} $$...

When I take derivatives in math I think of it as the amount of infinitesimals that change in one variable with respect to another, when the latter changes by one infinitesimal. But in physics those variables have real life meanings, so when I take the derivative of position with respect to time...

Folks, How do determine whether the derivative of a quadratic interpolation function ##ax^2+bx+c## is continous/discontinous in the context of the following We have a a true solution approximated by 2 quadratic interpolation functions ie, The approximation function f_1(x)=ax^2+bx+c...

Homework Statement Find the derivative of: 1. f(x)=arccos(5x^3) 2. f(x)=∫cos(5x)sin(5t)dt when the integral is from 0 to x Homework Equations Chain rule, dy/dx=dy/du*du/dx The Attempt at a Solution For the first one, I can just take 5x^3 as u and then apply the chain rule...

Hello,I am encountering some major confusion. When taking just garden variety f(x)=y derivatives of the form dy/dx, I don't encounter any problems. But recently I started taking derivatives of parametric equations, or switching things up using polar equations and I realized perhaps I'm not so...

Homework Statement Please look at the attached pic. I don't know how to type all these symbols in. Homework Equations Im not sure how to start The Attempt at a Solution I tried using the cyclic rule but the problem just started getting messier.

If a particle's position is given by x = 4-12t+3t^2 (where t is in seconds and x is in meters). a) What is the velocity at t = 1s? Ok, so I have an answer: v = dx/dt = -12 + 6t At t = 1, v = -12 + 6(1) = -6 m/s but my problem is I want to see the steps of using the formula v = dx/dt...

say you have a function f(x,y) \nablaf= \partialf/\partialx + \partialf/\partialy however when y is a function of x the situation is more complicated first off \partialf/\partialx = \partialf/\partialx +(\partialf/\partialy) (\partialy/\partialx) ( i wrote partial of y to x in case y was...

I'm given that x=cos3θ and that y=sin3θ if (d2y/dx2)=[(dy/dθ)/(dx/dθ)]/[dx/dθ] is right, wouldn´t the second derivative of the parametric be: 1/3c3θ ?? I got this by using dy/dθ=3sin2θ, and dx/dθ=-3cos2θsinθ any idea what's wrong? or is it right?

Find the first order partial derivatives of the function x = f(x,y) at the point (4,3) where: f(x,y)=ln|(x+√(x^2+y^2))/(x-√(x^2+y^2))| I understand the method of partial derivatives and implementing the given point values once the partial derivatives are found, however I am having trouble...

Homework Statement It says: \displaystyle f:{{\mathbb{R}}^{2}}\to \mathbb{R} \displaystyle f\left( x,y \right)=\left\{ \begin{align} & 1\text{ if 0<y<}{{\text{x}}^{2}} \\ & 0\text{ in other cases} \\ \end{align} \right. Show that all the directional derivatives about (0,0) exist but f...

I'm just experimenting with graphs of derivatives and have noticed something that our teacher has never adverted to but strikes me as very interesting: the third derivative when set to zero will give the inflection points of the first, which makes me reflect that the relationship which...

I am trying to understand the notion of a covariant derivative and Christoffel symbols. The proof I am looking at starts out with defining a tensor, Tmn = ∂Vm/∂xn where V is a covariant vector. I am having a mental block with regard to the indeces. How is it that the derivative of a...

A 3-dimensional graph has infinite number of derivatives (in different directions) at a single point. I've learned how to find the partial derivative with respect to x and y, simply taking y and x to be constant respectively. But what do I do if I want to take the partial derivative with respect...

Hi all, I have the following partial derivatives ∂f/∂x = cos(x)sin(x)-xy2 ∂f/∂y = y - yx2 I need to find the original function, f(x,y). I know that df = (∂f/∂x)dx + (∂f/∂y)dy and hence f(x,y) = ∫∂f/∂x dx + g(y) = -1/2(x2y2+cos2(x)) + g(y) Then to find g(y) I took the...

What I have learned in school is that differentiation and integration are opposites. By integrating a function we find the area under the graph. So, integration gives us the area. Differntiation gives slope of the function. If I am right by saying differentiation and integration are...

Homework Statement The problem is attached in the picture. The top part shows what is written in the book, but I am not sure how they got to (∂I/∂v)...The Attempt at a Solution It's pretty obvious in the final term that the integral is with respect to 't' while the differential is with...

Homework Statement What does it mean when the derivative of a function f(x) is in the form: d ln f(x) / d ln x ? Is it the logarithmic scale derivative, or something? Homework Equations d ln f(x) / d ln x The Attempt at a Solution Googling.

Hi. I am reviewing for an exam, and this is a topic that I did not go through very thoroughly. I understand how to calculate the derivative of an inverse function when I am given a point, as I simply use the equation (1/f((f^-1))'. So if I am given the equation y=x^3, for example, and am asked...

Homework Statement The equations xu^2 + yv = 2, 2yv^2 + xu = 3 define u(x,y) and v(x,y) in terms of x and y near the point (x,y) = (1,1) and (u,v) = (1,1). Compute the following partial derivatives: (A) ∂u/∂x(1,1) (B) ∂u/∂y(1,1) (C) ∂v/∂x(1,1) (D) ∂v/∂y(1,1) The answers are: (A)...

Please explain how this equation is derived. f'(x)= lim [f(x+h)-f(x)]/h h→0 Thanks.