Partial derivative Definition and 363 Threads

Homework Statement Given that the surface x^7y^2+y^4z^6+z^8x^8+9xyz=12 has the equation z=f(xy) in a neighbourhod of the point (1,1,1) with f(x,y) differentiable, find the derivatives. df/dx (1,1) = ? d^2f/dx^2 (1,1) = ? Homework EquationsThe Attempt at a Solution df/dx (1,1) I got -24/23 or...

Homework Statement a) Show that the function f(x,y)=\sqrt[3]{xy} is continuous and the partial derivatives f_x and f_y exist at the origin but the directional derivatives in all other directions do not exist b) Graph f near the origin and comment on how the graph confirms part (a). 2. The...

When can I do the following where ##h_{i}## is a function of ##(x_{1},...,x_{n})##? \frac{\partial}{\partial x_{k}}\frac{\partial f(h_{1},...,h_{n})}{\partial h_{m}}\overset{?}{=}\frac{\partial}{\partial h_{m}}\frac{\partial f(h_{1},...,h_{n})}{\partial x_{m}}\overset{\underbrace{chain\...

1. Problem Define a function: for t>=0, f(x,t) = { x for 0 <= x <= sqrt(t), -x + 2sqrt(t) for sqrt(t) <= x <= 2sqrt(t), 0 elsewhere} for t<0 f(x,t) = - f(x,|t|) Show that f is continuous in R^2. Show that f_t (x, 0) = 0 for all x. Then define g(t) = integral[f(x,t)dx] from -1 to 1. Show...

Homework Statement Find the partial derivative of ## w = xe^\frac {y}{z} ##. Homework Equations N/A The Attempt at a Solution ## \frac{∂f}{∂x} = e^y/z ## ## \frac{∂f}{∂y} = \frac{xe^y/z}{z} ## ## \frac{∂f}{∂z} = (-yz^-2)(xe^yz^-1) ## Are theses correct? Thanks everyone.

Homework Statement Find (∂z/∂x) of 6xyz Homework Equations N/a The Attempt at a Solution The correct answer is 6xy(∂z/∂x) but I would like proof of it. I got something different when I tried taking the partial derivative. 6xyz = 6x(yz) = Multiplication rule for derivatives 6(∂x/∂x) +...

Homework Statement Use the "mixed partials" check to see if the following differential equation is exact. If it is exact find a function F(x,y) whose differential, dF(xy) is the left hand side of the differential equation. That is, level curves F(xy)=C are solutions to the differential...

Hi, I'm using partial derivatives to calculate propagation of error. However, a bit rusty on my calculus. I'm trying to figure out the partial derivative with respect to L of the equation: 2pi*sqrt(L/g) (Yep, period of a pendulum). "g" is assumed to have no error. I know I can use the...

I've attached an image to this post. It essentially shows the equation for the first partial derivative using chain rule, which makes sense. What I'm confused with is how the second partial derivative was formulated. It seems they've simply squared the first partial derivative to find the second...

Homework Statement x^2 + y^2 < 1 Find the partial derivatives of the function. Homework Equations x^2 + y^2 < 1 The Attempt at a Solution @f/@x = 2x = 0 @f/@y = 2y = 0 4. Their solution @f/@x = 2x = 0 @f/@y = 2y + 1 = 0 5. My Problem I don't see how / why they get 2y + 1 for the...

I'm just doing some geometry and I can't remember how to find the maximum values of two variables in the same equation. Like, if you differentiate the equation partially for one, then again from the beginning partially for the other and add them together or something. The equation is something...

Homework Statement I want to find the partial derivatives in the point (0,0) of the function f:\mathbb R^2\rightarrow\mathbb R f(x,y):= \begin{cases} 0 & \text{if } (x,y) = (0,0) \\ \frac{y^5}{2x^4+y^4} & otherwise \end{cases} Homework Equations Our definition of the partial derivatives in...

Im doing a question on functionals and I have to use the Euler lagrange equation for a single function with a second derivative. My problem is I don't know how to evaluate \frac{d^2}{dx^2}(\frac{\partial F}{\partial y''}). Here y is a function of x, so y'=\frac{dy}{dx}. I know this is probably...

Homework Statement If w = w(x, y, z) is given implicitly by F(x, y, z, w) = 0, find a formula for both ∂w/∂z and ∂^2w/∂y∂z . You may assume that each function is sufficiently differentiable and anything you divide by during the process of your solution is non-zero. The Attempt at a Solution I...

Homework Statement Find the partial derivative of a*cos(xy)-y*sin(xy) with respect to y. Homework Equations None. The Attempt at a Solution The answer is -ax*sin(xy)-sin(xy)-xy*cos(xy). I know that I need to treat x as constant since I need to take the partial derivative with respect to y...

\mathcal{L}_M(g_{kn}) = -\frac{1}{4\mu{0}}g_{kj} g_{nl} F^{kn} F^{jl} \\ \frac{\partial{\mathcal{L}_M}}{\partial{g_{kn}}}=-\frac{1}{4\mu_0}F^{pq}F^{jl} \frac{\partial}{\partial{g_{kn}}}(g_{pj}g_{ql})=+\frac{1}{4\mu_0} F^{pq} F^{lj} 2 \delta^k_p \delta^n_j g_{ql} I need to know how...

I'm trying to come up with an expression for \partial y / \partial x where z = f(x,y). By observation (i.e. evaluating several sample functions), the following appears to be true: \begin{equation*} \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial...

Hello, we haven't really covered partial differentiation in my maths course yet, but it has come up a few times in mechanics where the 'grad' operator is being introduced, so I'm trying to learn about it myself. I'm looking at the partial derivatives section in "Mathematical Methods In The...

Homework Statement Find \frac{\partial f}{\partial x} if f(x,y)=\cos(\frac{x}{y}) and y=sinx Homework Equations See above The Attempt at a Solution For \frac{\partial f}{\partial x} I calculated -\frac{1}{y}\sin(\frac{x}{y}) which comes out as \frac{-\sin(\frac{x}{\sin(x)})}{sinx} and this...

Homework Statement Given the functions Q(v,w) and R(v,w) [/B] K = v(dQ/dv)r and L = v(dQ/dv)w Show that (1/v)K = (1/v)L + (dQ/dw)v (dW/dv)r I have the problem attached if for clarity of the information. Homework Equations I assume everything is given in the problem. The Attempt at...

Hi I need help regarding following can I write following partial derivative wrt x multiplied by Ax (∂A[x])Ax =∂(Ax^2)

Homework Statement Find (\frac{dV}{dp})_{n,T} for the Van de Waals gas law Homework Equations Van de Waals gas law: (\frac{p+an^2}{V^2})(V-nb)=nRT The Attempt at a Solution I just started doing problems like these so I would like to know if I am doing them right... What I did was I took...

Hi. Assume there's a probability ##q## for a guy to take a step to the right, and ##p=1-q## to take one to the left. Then the probability to take ##n## steps to the right out of ##N## trials is ##P(n) = {{N}\choose{n} }q^n p^{N-n}##. Now, what is ##<n>##? My textbook in statistical physics...

can anyone tell me the difference of application of total derivative and partial derivative in physics? i still can't figure it out after searching on the internet

many books only tell the operation of total derivative and partial derivative, so i now confuse the application of these two. when doing problem, when should i use total derivative and when should i use partial derivative. such a difference is detrimental when doing Physics problem, so i...

Homework Statement Given f(x, y, z) = 0, find the formula for (\frac{\partial y}{\partial x})_z Homework Equations Given a function f(x, y, z), the differential of f is df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy + \frac{\partial f}{\partial z}dz...

Homework Statement If xs^2 + yt^2 = 1 and x^2s + y^2t = xy - 4 , find \frac{\partial x}{\partial s}, \frac{\partial x}{\partial t}, \frac{\partial y}{\partial s}, \frac{\partial y}{\partial t} , at (x, y, s, t) = (1, -3, 2, -1) Homework Equations The Attempt at a Solution I...

I just need some clarification that this is fine so I have f_{x} = -2xe^{-x^2-y^2}cos(xy) -ysin(xy)e^{-x^2-y^2} now, taking the second derivative f_{xx} = [-2xe^{-x^2-y^2}+4x^2e^{-x^2-y^2}]cos(xy) - ysin(xy)[-2xe^{-x^2-y^2}]+2xe^{-x^2-y^2}sin(xy)y-cos(xy)e^{-x2-y^2}y^2

I've seen it written two different ways: $$\frac{\partial f}{\partial x} = \lim\limits_{h \rightarrow 0} \frac{f(x + h, y) - f(x,y)}{h}$$ and $$\frac{\partial f}{\partial x} = \lim\limits_{h \rightarrow 0} \frac{f(x_0 + h, y_0) - f(x_0,y_0)}{h}$$ where the latter evaluates the function at...

if U = sin^-1 (x/y) +cos(y/x) then Ux/Uy = ? ans is -y/x. i have specially doubt on the partial derivative of U w.r.t y for inverse sin. thank you in advance

I've come across using partial derivative notation for taking the partial derivative of a function f with respect to a vector x. I've never seen this before. It is also being referred to as a gradient. However, I have only seen gradients where all variables in the space are featured in the...

In a problem that requires converting from cartesian to polar coordinates, I need to take \frac{dr}{dx}. I tried doing it two different ways but getting two completely different answers.. Method 1: r=\sqrt{x^2+y^2} \frac{dr}{dx}=\frac{1}{2}\frac{1}{\sqrt{x^2+y^2}}2x \;\; =...

Hello, Wolfram is giving me the required answer however, the steps it uses I find very confusing. Can anyone share some light on how wolfram achieved the correct answer. As I am new to this site, I won't be using any code. I am in the process of writing it up on Latex. Here is the link...

Homework Statement (50 - x - y)(x+y) - x - ((y^2)/2)) simplified: 50X - X^2 - XY - 50Y - XY - Y^2 - X - ((Y^2)/2)Homework Equations 50X - X^2 - XY - 50Y - XY - Y^2 - X - ((Y^2)/2)) The Attempt at a Solution for x: 49 - 2x - 2y for y: 50 - 2x - 3y The book has the same, except -1x and...

Hi! :smile: I have the following integral \int^{∞}_{∞} \frac{\delta^{n}}{\delta a}f(a,b,c)da there is any way to rewrite it in terms of: \int^{∞}_{∞} f(a,b,c)da I want to evaluate it for the case of n=1,2 and 3. Thanks you so much.

I am not quite sure how \frac{\partial}{\partial u}\left(\frac{\partial z}{\partial u}\right) =\frac{\partial}{\partial u}\left( u \frac{\partial z}{\partial x}+v\frac{\partial z}{\partial y} \right) comes to \frac{\partial z}{\partial x} + u\frac{\partial}{\partial u}\left(\frac{\partial...

Homework Statement If possible, please check my work for any large errors. y = 10kl - √k - √l k = (t/5) + 5 l = 5e^t/10 Evaluate at t = 0 using chain rule. Homework Equations y = 10kl - √k - √l k = (t/5) + 5 l = 5e^t/10 The Attempt at a Solution = ∂y/∂k * dk/dt + ∂y/∂l * dl/dt = (10l -...

Homework Statement A function f(x,t) depends on position x and time t independent variables. And if \dot{f} represents \frac{df(x,t)}{dt} and \dot{x} represents \frac{dx}{dt}, then find the value of \frac{\partial\dot{f}}{\partial\dot{x}}. Homework Equations The Attempt at...

Homework Statement Find an equation for the path of a particle that starts at P(10,10) and always moves in the direction of maximum temperature increase if the temperature in the plane is T(x,y) = 400-2x^2 -y^2 Homework Equations T(x,y) = 400-2x^2 -y^2 dT/dx = -4x dT/dy = -2y...

Hi everyone! I'm not sure if this is the right forum to post my question. If I'm wrong, let me know it. The question: Let us consider the functions \theta=\theta(x,y), and M=M(\theta), where M is a operator, but i doesn't relevant to the problem. I need to know the derivative \frac{\partial...

Homework Statement Hi Say I have a function f(x(t), t). I am not 100% sure of the difference between \frac{df}{dt} and \frac{\partial f}{\partial t} Is it correct that the relation between these two is (from the chain rule) \frac{df}{dt} = \frac{\partial f}{\partial t} +...

For spherical coordinates, u(r,\theta,\phi) is function of r,\theta,\phi. a is constant and is the radius of the spherical region. Is: \int_{0}^{2\pi}\int_{0}^{\pi}\frac{\partial\;u(r,\theta,\phi)}{\partial {r}}a^2\sin\theta d\theta d\phi=\frac{\partial}{\partial...

I was working on a pde, and I needed to compute a Jacobian for it. Suppose we have a function consisting of a series of matrices multiplied by a vector: f(X) = A * B * b --where X is a vector containing elements that are contained within A, b, and/or b, --A is a matrix, B is a matrix, and b is...

Hello Everyone, So in other words, if you didn't understand what I'm saying from the title of this post, look at it this way: What is the answer to this integral? ∫(partial dx)/(partial dt) * dx According to my textbook the answer is 0 but I'm getting easily confused as to how this is...

Hi all, I've got a question regarding a price elasticity problem and a partial derivative. That's what's given for the exercise: So, first of all we calculate all the demand with the given information. Which is: And then we come to the actual problem. (4. b) ) How do they...

Homework Statement Let W = F(u(s,t),v(s,t)) (in my notation, u_s would represent du/ds u(1,0) = -7 v(1,0) = 3 u_s(1,0)=8 v_s(1,0)=5 u_t(1,0)=-2 v_t(1,0)=-4 F_u(-7,3)=-8 F_v(-7,3)=-2 Find W_s(1,0) and W_v(1,0) Sort of having a hard time getting started here... I believe...

could someone show me how \frac{∂}{∂t}(\frac{∂z}{∂u})= \frac{∂^2z}{∂u^2} \frac{∂u}{∂t} where z=z(u) u=x+at

Homework Statement The function given is (1+xz)^(1/2) + (1-xy)^(1/2) I have to take the partial derivative with respect to x, y, and z. The question says Choose the order wisely. I don't understand what it means? How could I choose the order badly? Can anyone skilled in explaining math to...

Homework Statement \frac{\partial f}{\partial t},\frac{\partial f}{\partial x} where f=f(x,t,\frac{dx}{dt}) Homework Equations The Attempt at a Solution I think it's impossible to consider it as a simple partial derivative.

Homework Statement Find the first partial derivatives ∂z/∂x and ∂z/∂y of sin(0x+5y+z)=0 at (0,0,0). Homework Equations sin(0x+5y+z)=0 The Attempt at a Solution 0x+5y+z=kπ z=kπ-5y So, ∂z/∂x= 0 and ∂z/∂y= -5 What I do not understand is WHY 0x+5y+z=kπ is an acceptable...