Derivatives Definition and 1000 Threads

Homework Statement Suppose that the equation F(x,y,z) = 0 implicitly defines each of the three variables x, y and z as functions of the other two: z = f(x,y), y = g(x,z), z = h(y,z). If F is differentiable and Fx, Fy and Fz are all nonzero, show that \frac{∂z}{∂x} \frac{∂x}{∂y} \frac{∂y}{∂z} =...

Homework Statement The acceleration of a body is defined as a=-K*u^2 , K is const. When t=o sec V=Vo. Find : a) V(t) b) X(t) c) V(x) . Homework Equations The Attempt at a Solution

Let a represent the area, p the perimeter, d the diagonal, b the breadth, and L the length of a rectangle. One can easily write down from analytical geometry all the various relationships between the above variables, and from these obtain directly a variety of partial differential quantities...

Homework Statement \frac{d}{dx}e^{ax^{3}} I'm simply trying to determine whether or not I am doing these correctly and applying the chain rule properly. Homework Equations Chain rule et al. The Attempt at a Solution \frac{d}{dx}e^{ax^{3}} e^{ax^{3}}\frac{d}{dx}ax^{3} e^{ax^{3}}a(3)x^{2}...

Homework Statement I was messing around online when I found this: \frac{dy}{2} = 2x. This was derived from the function y = x2. I had never really seen anything like this before. When I solved for "dy," I got 4x. However, for example, when x changes from 0 to 2, the y changes from 0 to 4...

Homework Statement If V=xf(u) and u=y/x, show that x^2.d2V/dx2 + 2xy.d2V/dxdy + y^2.d2V/dy2= 0 (This a partial differentiation problem so all the d's are curly d's) The Attempt at a Solution I have tried to work out d2V/dx2 and the other derivatives, then multiply them by x^2 or 2xy or...

Hi, Do you know the name of this kind of singularity at A ? The function is finite but the left derivative is +\infty and the right derivative is -\infty. http://shareimage.org/viewer.php?file=mt79897bbpxxse1v8pzb.jpg Thanks

Dear Everybody! I'm searching for some real life applications of partial derivatives. I would be very thankful, if you sent me some example. Thanks from Hungary.

I have come to a bit of a misunderstanding with partial derivatives. I will try to illustrate my problem. Say we have a function f(x, y(x), y'(x)) where y'(x)=dy/dx. Now suppose that f does not explicitly depend on x. My physics book says at this point that ∂f/∂x=0, even though y(x) and y'(x)...

Homework Statement Let f = f(u,v) where u = x+y , v = x-y Find f_{xx} and f_{yy} in terms of f_u, f_v, f_{uu}, f_{vv}, f_{uv} Then express the wave equation \frac{\partial^2f}{\partial x^2} - \frac{\partial^2f}{\partial y^2} = 0 Homework Equations Chain rule, product rule...

What is the difference? I was pretty bored last night so I got onto Yahoo Answers and answered a few calculus questions. It was a simple integration by parts question: \intxsin(x) dx I solved as: u = x du = dx dv = sin(x) dx v = -cos(x) uv - \intvdu -xcos(x) + \intcos(x)dx =...

Homework Statement y''+(1/y)*(y')2=0 Homework Equations The Attempt at a Solution This is another problem I am having trouble with. I have done searches around the internet, but seen that all "non linear" ODE of second order involves a non linear form in a non differential term...

URGENT !Year 11 Double variable derivatives I am having trouble with this question it is derivatives. Previously I have been able to complete these with no trouble but am a little confused with how start this one: y= a^2(3x+5)^3. I don't know whether to use the product rule and just leave...

Why can you do this? thanks!

let C0 be the set of continuous functions f : R -> R. For n >= 1, let Cn denote theset of functions f : R -> R such that f is differentiable and such that f' is contained in C(n-1). (Therefore Cn is the set of functions whose derivatives f',f'',f''',...,f^(n) up to the nth order exist and are...

Hello, this question will essentially concern quantum field theory in curved spacetime, and it has two parts to it. I have recently acquired DeWitt's treatment of the formalism, which immediately discusses the role of killing vectors in the theory. Specifically, given a killing vector field...

Hope this is the right section. I'm having trouble ironing out an apparent inconsistency in matrix trace derivative rules. Two particular rules for matrix trace derivatives are \frac{\partial}{\partial\mathbf{X}} Tr(\mathbf{X}^2\mathbf{A})=(\mathbf{X} \mathbf{A}+\mathbf{A} \mathbf{X})^T...

Homework Statement Find all second partial derivatives of z=arctan((x+y)/(1-xy))Homework Equations d/dx of arctan(x) is 1/(1+x^2)The Attempt at a Solution Not sure how to proceed... I don't want the answer, just an idea as to how to move forward. My attempt at finding the first...

Homework Statement Find the Derivative of: (ln(cos4x)) / 12x^2 Homework Equations y' ln(x) = 1/x The Attempt at a Solution I have determined the correct answer, but I am still confused as to how I came to the solution. Starting with the numerator, the derivative of cos...

Homework Statement A man 6 feet tall walks away from a streetlight that is 18 feet tall. If the length of his shadow is changing at a rate of 3 feet per second when he is 25 feet away from the base of the light, how fast is he walking away from the light at this moment? Homework Equations...

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Let \varphi be a one-parameter group on a manifold M, and let f be a differentiable function on M, the derivative of f with respect to \varphi is the defined as the limit: \lim_{t\to 0} \frac{\varphi^*_t[f]-f}{t}(x)=\lim_{t\to 0}\frac{f\circ \varphi_x(t)-f\circ...

Hi. Doing a bit of self study. I would like to know how to understand the derivative. I understand the algebra and procedural stuff that you need to do to get the derivative of a function. Is there a way I can understand it graphically? Say I draw y=x^2 on a graph. Then I draw y=2x on...

Homework Statement 1. The line perpendicular to the curve y = 2x^3 - x^2 + x - 3 at the point (1, -1) will intersect the x-axis at what point? 2. f(x) = |x^2 - 5| - x, for all x. Let g = f(f(f(x))), find g'(2). I tried just subbing in 2x - 1, the first derivative, to f(2x - 1) and...

Homework Statement Let f(x) = axe^((bx)^2). Find the value for a times b if it's known that there's a max value of 2 at x = 3. Second, There is one line which is tangent to the curve y = 1/x, at some point A and at the same time tangent to the curve x^2 at some point B. What is the...

Homework Statement Let's say I have a function for a circle x^2 + y^2 = C where C is a constant. Then this is a cylinder with the z-axis. Now in my ODE book, we would normally define it as F(x,y) = C = x^2 + y^2 as a level surface. Now my question is about what the partial...

Homework Statement if f is infinitely continuously differentiable and f(0) = 0 then prove that all derivatives of f at 0 are 0 iff lim f(x)/x^n = 0 as x --> 0 Homework Equations The Attempt at a Solution I didnt know whether to use induction on this, I tried a base case so...

Homework Statement Find the general function f(x,y) that satisifes the following first-order partial differential equations \frac{df}{dx}=4x^3 - 4xy^2 + cos(x) \frac{df}{dy}=-4yx^2 + 4y^3 The Attempt at a Solution I integrated both to get: x^4 - 2x^2y^2 + sin(x) + c(y) and -2y^2x^2 + y^4...

Homework Statement The position on the ground in the xy plane that is hit by the sun given by (x,y)=(3t+tan(phi), -2t+tan(theta)), where t, phi, and theta, are controlled input variables. What is the velocity of the hit point if the input variables are at values (5, pi/4, pi/3) and changing...

Something has been bugging as of late: usually, derivatives (ordinary and partial) are defined for interior points. However, I often come across statements in which they seem to also be defined for boundary points. For example, Leibniz' rule of integration, as usually stated, assumes some...

Homework Statement If f(x) can be differentiated, find expressions for the derivatives of the following functions. a) g(x) = f(x6) b) h(x) = [ f(x)]6 c) f(x) = x2/ f(x) The Attempt at a Solution a) b) Use the product rule first then multiply that expression by the expression for...

Lee: Introduction to Smooth Manifolds, definition A.18: He then shows, by the chain rule, that D_vf(a_0)= \sum_{i=1}^n v^i \frac{\partial }{\partial x^i}f(a) \bigg|_{a_0} It seems to me, though, that this number depends not only on the direction of v but also on its length. For example...

I know that Mathcad only takes partial derivatives. I set up my equations using this general format: L:= x(t) Then, I take the derivative of L with respect to t and get the following: dL/dt -> d/dt*x(t) However, when I take the derivative of L with respect to x, I should get 1, but...

I want to interpolate a function between the points A, B, C. At A and C I only know the value of the function, but at B (lying between them) I also know the function's first and second derivatives. How would you interpolate between these points?

Dear PF, I am having very heavy supersymmetric derivatives to do (and in general heavy calculations) could you advise me any software/package or something that will help me to do my calculations on computer? Thank you very much Rgds GT

Homework Statement Use Clairaut's Theorem to show that is the third order partial derivatives are continuous, then fxxy=fyxy=fyyz Clairaut's Theorem being: fxy(a,b)=fyx(a.b) Homework Equations fxyy=d/dy(d2f/dydx)=d^3f/dy^2dx The Attempt at a Solution Tried to differentiate...

Homework Statement See attachment. 2. Homework Equations /solution attempt Part (a) Well, the gradient evaluated at (1,2-1) will give the rate of change. If we want the maximum rate of change then we need the directional direction such that the unit vector \mathbf{u} is in the same...

hi does anyone know why the 2nd derivative chain rule is as such? i roughly know that if u = f(x,y) and x=rcos(T) , y = rsin(T) then du/dr = df/dx * dx/dr + df/dy * dy/dr but if i am going to have a second d/dr, then how does it work out?

hi i have a problem for this if f(x-z)=x+y+z solve can i say u=x-z and write F(x,y,z)=x+y+z-f(u) and then or this isn't true ? thanks if u help me.

find the directions in which the directional derivative of f(x,y)=x^2 + sin(xy) has a value of 1 at the point (1,0) Fx=2x+ycos(xy)=2 Fy=xcos(xy)=1 So we have <2,1> and we need to find vectors that dotted with <2,1> =1 <2,1>.<x1,x2>=1 2x1+x2=1 So whn x1 is 0 we have x2 is 1 so...

Hello! My problem is that I want to find (\frac{\partial}{{\partial}x}, \frac{\partial}{{\partial}y}, \frac{\partial}{{\partial}z}) in spherical coordinates. The way I am thinking to do this is...

Homework Statement Everyline tanjent to the function y=sin x has a y-intercept. Among all these tanjent lines, somewere between 0<x<2pi, find the equation of the line with the highest y-intercept. Homework Equations derivative of sinx=cosx Second derivative is -sinx The Attempt at a...

Homework Statement function f is differentiable when x=0, f'(0) is not equal to zero for all real Numbers f(a+b) = f(a)f(b) show f'(x) = f'(0)f(x)Homework Equations The Attempt at a Solution f(x+0) = f(x) = f(x)f(0) this shows f(0) = 1 then i get stuck..

Homework Statement function f is differential when x=0, f'(0) is not equal to zero for all a,b(real Numbers) f(a+b) = f(a)f(b) show f'(x) = f'(x)f(x) Homework Equations The Attempt at a Solution f(a+b) = f(a)f(b) for all a,b(real numbers) f(0), a+b=0 then f(0) =...

Suppose u(x) is periodic with period 2\pi. Also m\le u(x)\le M. Then is it possible for some derivatives of u(x) to be outside [m, M]? In other words, can any derivative be 2pi-periodic and have a different amplitude? When u(x) is a sine curve, then it is not true because, the frequency of...

Hello all, This is a question on a problem set for my Calculus 1 class. Please help if you can. Homework Statement A function, f defined on the set of real numbers is said to have even symmetry if f(-x) = f(x) for all x, and is said to have odd symmetry if f(-x) = -f(x) for all x. Use...

Homework Statement http://www.math.wvu.edu/~hjlai/Teaching/Tip-Pdf/Tip3-27.pdf Example 7. Not this question in particular, but it shows what I'm talking about. I understand how they get the first partial derivative, but I'm completely lost as how to take a second one. I have tried...

I recently read a paper on fractional derivatives. That is how to take derivatives of fractional order rather than the usual integral order. The paper made perfect sense to me, however I wondered: 1) Are there geometric interpretations of fractional derivatives? Kind of like how first...

I started to answer this question, and I have quite a bit an answer, but still not complete, let's say that we write a Lagrangian in QFT, which an unknown function of the scalar field \phi and its derivative \partial \phi. We can always Taylor-expand it and get: L(\phi,\partial\phi) = a + b \phi...

Just a quick question... if we have f(x,y,z) and x(t), y(t), z(t), without substituting in what x y and z are in f, how do we calculate df/dt?