Partial derivatives Definition and 417 Threads

1. In the Khan academy video I watched on partial derivatives, I understand absolutely everything except for the last 20 seconds which confused me. http://www.youtube.com/watch?v=1CMDS4-PKKQ Using the formula: Z = x² + xy + y² @z/@x = 2x +y x=0.2, y=0.3 2(.2) + .3 = .7 What...

We have a function f:R^2->R and it has partial derivative of 2nd order. Show that f_{xy}=0 \forall (x,y)\in \mathbb{R}^2 \Leftrightarrow f(x,y)=g(x)+h(y) The <= is self explanatory, the => I am not sure I got the right reasoning. I mean we know that from the above we have: f_x=F(x) (it's...

Homework Statement Find the partial derivatives: f(x,y)= integral[x,y] cos(t^2)dt, find f_x(x,y) and f_y(x,y) Homework Equations I know from calculus that the derivative of an integral is the function. The Attempt at a Solution I found that the integral of [x to y]...

Homework Statement The question is attached along with its solution. Homework Equations Partial differentiation and the implicit function theorem. The Attempt at a Solution My work is attached. I feel it's correct but is it incomplete? I have the following questions/confusions...

Homework Statement In the steps below, the ∂z/∂x does not seem to be obeying normal algebraic rules. I'm confused. This is not really a problem, I'm just trying to understand the steps. The Attempt at a Solution 1. 3z2∂z/∂x - y + y∂z/∂x = 0 2. ∂z/∂x = y/(y + 3z2) if ∂z/∂x were...

If you have a function f(x,y)=xy where y is a function of x, say y=x^2 then the partial derivative of f with respect to x is \frac{\partial f}{\partial x}=y However, if you substitute in y and express f as f(x)=x^3 then the partial derivative is \frac{\partial...

Homework Statement ∂f/∂x (xy -1)2 = 2y(xy-1) The Attempt at a Solution I would think the answer would be 2(xy-1) I don't understand where the y comes from in 2y

Homework Statement I don't understand why ∂f/∂x = xy = y whereas ∂f/∂x = x2 + y2 = 2x Why does the y disappear in the second but not in the first?

Homework Statement Suppose the function f:R^2→R has 1st order partial derivatives and that δf(x,y)/δx = δf(x,y)/δy = 0 for all (x,y) in R^2. Prove that f is constant; there exists c such that f(x,y) = c for all (x,y) in R. There's a hint as well: First show that the restriction of...

I was reading a section on vector fields and realized I am confused about how to take partials of vector quantities. If V(x,y)= yi -xj, I don't understand why the \partialx= y and the \partialy= -x. The problem is showing why the previous equation is not a gradient vector field (because the...

If a and b are constants, compute the expression KY'(K) + LY'(L) for Y = AK^a + BL^a Y'(K) means partial derivative with respect to K by the way. The answer in the book is KY'(K) + LY'(L) = aY I'm not sure what they did or what they're asking :/

Homework Statement I want to show that the partials exist for a certain function. Homework Equations My book says that if a function f is differentiable at a point x then the partial derivatives exist. The Attempt at a Solution Rather than showing f is differentiable, I am...

In this textbook, how exactly are 2.2.19 and 2.3.21 equal? http://i.imgur.com/LrcXE.png

Hi, I remember having read in basic calculus that the following is true, but I don't know what this property is called and am having a hard time finding a reference to this. d u(x,y) = \frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy Ques: Is this true ? Is this true for...

I know that for any C2 function, the mixed second-order partials are equal, and I see that this should extend inductively to a statement about the kth partials of a Ck function, but I am having trouble figuring out exactly how this works. For example, take f:ℝ2 → ℝ . fxxy=fxyy is not true...

Hi there, just wanted to make a clarification before my final exam. The second derivative test for partial derivatives (or at least part of it) states if D = ∂2f/∂x2 * ∂2f/∂y2 - (∂2f/∂x∂y)2 and (a,b) is a critical point of f, then a) if D(a,b) > 0 and ∂2f/∂x2 < 0, then there is a local...

Homework Statement Find (x,y) which maximizes f(x,y) for x ≥ 0. f(x,y) = e-x - e-2x + (1 - e-x)(4/5 - (3/4 - y)2)Homework Equations The Attempt at a Solution Due to the question prior to this one, I know all the first order and second order partial derivatives of the formula. I do not...

Homework Statement When two resistors R1 and R2 are connected in parallel, their effective resistance R = (R1R2)/(R1+R2). Show that is R1 and R2 are both increased by a small percentage c, then the percentage increase of R is also c. Homework Equations The Attempt at a Solution I...

Homework Statement Let F(x,y) be a twice differentiable function such that 4 * Fx2 + Fy2 = 0. Set x = u2 - v2 and y = u*v. Show that Fu2 + Fv2 = 0The Attempt at a Solution Fu = Fx*2u + Fy*v Fv = Fx*-2v + Fy*u Fu2 = 4v2Fx2-4FxFy*u*v+Fy2u2 Fv2 = 4u2Fx2+4FxFy*u*v+Fy2v2 Adding these two...

Homework Statement So there is an exercise in which I should "verify" the chain rule for some functions. In other words to do it by substitution, then doing by the formula and checking if the results are the same. (and checking with the book`s answer too) For a few of them, they just don`t...

Hi, in general can we tell anything about the partial derivatives of a differentiable function if they are equal? for example I would like them to have to equal some constant. Would this be true?

Homework Statement Consider the system of equations x^2y+za+b^2=1 y^3z+x-ab=0 xb+ya+xyz=-1 1. Can the system be solved for x, y, z as functions of a and b near the point (x, y, z, a, b)=(-1, 1, 1, 0, 0)? 2. Find \frac{\partial x}{\partial a} where x=x(a, b) The Attempt at a...

Homework Statement I need to prove that x\frac{ \partial^2z}{ \partial x^2} + y\frac{\partial^2z}{\partial y\partial x} = 2\frac{\partial z}{\partial x} Homework Equations z = \frac{x^2y^2}{x+y} The Attempt at a Solution I actually did it the long way and I got the right answer...

The question I'm trying to solve is part (ii) of the attached file I've used partial derivatives to input back into the Black Scholes equations and after factorising it, I've got it down to: (a + 2bt + αt +r) * (S².c.e^(at+bt²) = 0 I'm now stuck on what to do next, as there would need to be...

Homework Statement Given F(x_1,x_2,...,x_i,...,x_n) = nth-root(x_1*x_2*...*x_i*...*x_n), how do I take the partial derivative with respect to x_i, where x_i is an arbitrary variable? Homework Equations The Attempt at a Solution Would it just be...

1. The problem statement, all variables and given known data Find the partial derivatives (1st order) of this function: ln((\sqrt{(x^2+y^2} - x)/(\sqrt{x^2+y^2} + x)) Homework Equations The Attempt at a Solution I obviously separated the logarithm quotient into a...

Homework Statement A) if http://www4d.wolframalpha.com/Calculate/MSP/MSP10819hf7feh5hf5hh7e00004ga728h1d42i6h0c?MSPStoreType=image/gif&s=18&w=161&h=20 Find the partial derivatives? B) Suppose you are manufacturing blackboards and whiteboards and that P is your monthly profit when your...

Homework Statement This is a bonus problem on our homework, and I'm having trouble figuring out how to setup what I need. Homework Equations Here are my best guesses: f_x=\frac{\partial f}{\partial x} f_y=\frac{\partial f}{\partial y} f_{xx}=\frac{\partial}{\partial...

In James Stewart's Calculus exercise 82 page 891 asks you to show that: \frac{\partial P}{\partial V}\frac{\partial V}{\partial T}\frac{\partial T}{\partial P} = -1 I can do this by noting that V = \frac{nRT}{P} so that: \frac{\partial V}{\partial T} = \frac{\partial}{\partial...

Homework Statement Suppose f(x,y) = 2x^5 + 4xy + 2y^3 g1(u,v) = u^2 - v^2 g2(u,v) = uv h(u,v) = f(g1(u,v), g2(u,v)) Use chain rule to calculate: dh/du (1,-1) and dh/dv (1,-1) Homework Equations The Attempt at a Solution i let h (u,v) = 2(u^2 - v^2)^5 + 4(u^2-v^2)(uv) +...

Homework Statement Hi I'm just looking for a link to the proof of this theorem: if the partial derivatives of function f exist and are continuous at a point then the function is differentiable there Or even the name would be helpful Its not a homework assignment per say, just something...

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} =...

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 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...

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...

Why can you do this? thanks!

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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 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 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...

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.

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 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...

please help about this if f(u,v)=f(y/x,z/x)=0 and z=g(x,y) and show thanks alot

Homework Statement Let z = z (x,y) be a function with x = x(s), y = y(t) satisfying the partial differential equation (Ill write ddz/ddt for the partial derivative of z wrt t and dz/dt for the total derivative of z wrt t, as I have no idea how to use Latex.) ddz/ddt +...

Homework Statement Sorry I tried to use Latex but it didn't work out :/ Make the change of variables r = x + vt and s = x vt in the wave equation partial^2y/partialx^2-(1/v^2)(partial^2y/partialt^2)=0 Homework Equations...

Domain of f(x,g(x)), and partial derivatives Watching http://www.khanacademy.org/video/exact-equations-intuition-1--proofy?playlist=Differential%20Equations Khan Academy video on exact equations, I got to wondering: if x is a real number, what is the domain of a function defined by f(x,g(x))...

Homework Statement Heat is being conducted radially through a cylindrical pipe. The temperature at a radius r is T(r). In Cartesian co-ordinates, r = \sqrt{(x^{2}+ y^{2}}) show that \frac{\partial T}{\partial x} = \frac{x}{r} \frac{dT}{dr}