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Questions tagged [partial-derivative]

For questions regarding partial derivatives. The partial derivative of a function of several variables is the derivative of the function with respect to one of those variables, with all others held constant.

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The equality of all mixed $s$th-order partial derivatives

As we know, $ f(x,y)={\frac {x^{3}y-xy^{3}}{x^{2}+y^{2}}}$ , if $(x,y) \ne (0,0)$ and $f(0,0)=0$.All of its second-order partial derivatives exist on $ \mathbb R^2$. But $\frac{\partial ^{2}}{\...
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1answer
24 views

Taylor in two variables: Can we know that two functions are different.

Consider the following setup: Two functions $f,g:\mathbb{R}^2\to\mathbb{R}$ that are twice continuously differentiable such that: $f(0,0)=g(0,0)$ $f_x(0,0)=g_x(0,0)$ $f_y(0,0)\neq g_y(0,0)$ Can we ...
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1answer
28 views

Derivate of vector : transpose, conjugate and conjugate transpose

Let $x$ and $y\in \mathbb{C}^{K\times 1}$ and $H\in \mathbb{C}^{K\times K}$ a diagonal matrix. $\bar{x}$ denotes the complex conjugated, $x^{T}$ denotes the transpose and $x^{*}$ denotes the complex ...
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1answer
49 views

Find partial derivatives of $f(x,y)= \sqrt {xy}$ at the point $(0,0)$

I'm trying to find partial derivatives $f_x$ and $f_y$ at $(0,0)$, assuming they exist (although I believe they don't), of the function $f(x,y)= \sqrt {xy}$ when I use the chain rule, I get partial ...
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1answer
28 views

If $f: U \to \mathbb{R}$ has bounded partial derivatives in $U$, then $f$ is continuous on $U$.

Problem. Let $U \subset \mathbb{R}^{m}$ be an open. Show that if $f: U \to \mathbb{R}$ has bounded partial derivatives in $U$, then $f$ is continuous on $U$. Idea. $$\lim_{\lambda \to 0}|f(x + \...
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Solve this problem and proper explain this problems focus on partial deffrential equation and to prepare pde [on hold]

A non trivial solution of boundary value problemand monotonically increasing function
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1answer
30 views

Not sure that this function isn't differentiable at the origin

For a fixed $k\in\mathbb{N}$, define $f_k\colon\mathbb{R}^2\to\mathbb{R}$ by: $$f_k(x,y) = > \begin{cases}f_k(x,y)=\frac{x^2(x+y^2)}{x^2+y^{2k}}\mbox{ > if}~(x,y)\neq(0,0)\\0\mbox{ if} ~(x,y)=(...
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2answers
57 views

Let $z=(x)^y$, what is $\frac{ \partial ^2z}{\partial x \partial y}$?

Let $z=x^y$, what is $\displaystyle \frac{\partial ^2z}{\partial x \partial y}$ ? My answer is $ \displaystyle \frac{ \partial z}{ \partial y} = x^y \ln x$ $ \displaystyle \frac{ \partial ^2z}{\...
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1answer
28 views

Partial Differentiation: Suppose $f(r,\theta,\phi)$ and $x=r\sin(\theta)\cos(\phi)$. How to find $∂f/∂x$?

Partial Differentiation: Suppose $f(r,\theta,\phi)$ and $x=r\sin(\theta)\cos(\phi)$. How to find $∂f/∂x$? I have the following question and no access to solutions. The variables $x$, $y$, $z$ and $r$...
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1answer
20 views

Partial derivatives of this function

Could someone please confirm for me the partial derivatives of this function: $\dfrac{\mathrm{\partial}L}{\mathrm{\partial}U}$ and $\dfrac{\mathrm{\partial}L}{\mathrm{\partial}V}$: $L = \dfrac{1}{N} \...
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2answers
45 views

Partial derivative of $xy\frac{x^2-y^2}{x^2+y^2}$ [duplicate]

I am asked to show, that $f(x,y)=\begin{cases} xy\frac{x^2-y^2}{x^2+y^2}\space\text{for}\, (x,y)\neq (0,0)\\ 0\space\text{for}\, (x,y)=(0,0)\end{cases}$ is everywhere two times partial ...
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0answers
32 views

derivative of $[n \times \nabla \eta]^2$ with respect to $\eta$

I trying to implement an equation numerically, Numerically I can do it in, but was wondering if there is an analytical solution to it. Here is the equation: $ f = [n \times \nabla \eta]^2$ Where, $...
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1answer
17 views

If $A$ is convex and $f$ has first partial derivative zero, then $f(x',y) = f(x,y)$ for every $(x',y),(x,y) \in A$?

Let $A \subset \mathbb{R}^{2}$ be open and $f:A \to \mathbb{R}$ of class $C^{1}$. Suppose that $D_{1}f(a) = 0$ for every $a \in A$. (a) If $A$ is convex, then $f(x',y) = f(x,y)$ for every $(x',y),...
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1answer
21 views

Application of chain rule for partial derivatives

A rectangular metal block above has length y com and a square cross section of side x cm. When the metal block is heated, the area of cross-section A and the length of the metal block increase at a ...
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3answers
45 views

I can't solve this basic equation?

I need to make either $y$ or $x$ the subject of this equation but I can't seem to get either one $\frac{2x}{1+x^2+y^2}+ \frac{1}{\sqrt2} =0$ I always end up with $-2\sqrt2x-1-x^2=y^2$ which is ...
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1answer
22 views

finding $\frac{dz}{dt}$ of $z(x,y)= x^2y^3$ , $x(t)= 2t^3$ , $y(t) = 3t^2$

finding $\frac{dz}{dt}$ of $z(x,y)= x^2y^3$ , $x(t)= 2t^3$ , $y(t) = 3t^2$ First I found $\frac{\partial z}{\partial x} = 2xy^3$ $\frac{\partial z}{\partial y} = 3 x^2 y^2$ $ x’(t) = 6t^2 $ $y’(...
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3answers
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Partial derivative of $e^{x^2 + 2y^2 + 3z^2}$

Find $f_y$ of $e^{x^2 + 2y^2 + 3z^2}$ How do I do this ? I am trying to follow the formula of - $\frac{d}{dx} e^{ax +b} = ae^{ax+b} $ Since I am differentiating the independent variable of y, ...
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1answer
49 views

'any sequence of $n$ partial differentiations of $f$ results in a constant times $f^{n}$'

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch5.1 In the last part of Ch5.1 of the text, just before a corollary (Cor 5.5), it says: '$\...
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0answers
39 views

pdf of a function of a normal random variable

Let $f$ be a function $f : x \mapsto y$ where $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$; $m \geq n$. $f$ is not invertible. I have a random variable $X$ s.t. $X \sim \mathcal{N}(\mu, \Sigma)$ ...
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2answers
129 views

What is meant by $\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=-1$ ? How to interpret it?

Let $F(x,y,z)=0$. So $x,y,z$ are defined implicitly in function of the other variable, i.e. $x=x(y,z)$, $y=y(x,z)$ and $z=z(x,y)$. Now $$dx=\frac{\partial x}{\partial y}dy+\frac{\partial x}{\partial z}...
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2answers
38 views

Differentiation Under the Integral Sign with Variable Substitution

Let $\psi(x,\xi)$ take inputs $x = (x_1,...,x_n),\ \xi = (\xi_1,...,\xi_n)\in\mathbb{R}^n$ and let $\psi$ have continuous first partial derivatives. I wish to show that $$ \frac{\partial}{\partial ...
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1answer
81 views

Computing Neural Network Gradients

The following note "Computing Neural Network Gradients" explains how we can take derivate with respect to matrix and vector. I have some questions: Figure below from the above note shows when we take ...
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38 views

Proving for $u=f(x,y), v=g(x,y)$, $\frac{du}{dx}\frac{dx}{du}+\frac{dv}{dx}\frac{dx}{dv}=1$.

How do you prove: If u and v are differentiable, $$u=f(x,y), v=g(x,y)$$ Prove $$\frac{\partial u}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial v}{\partial x}\frac{\partial x}{\partial v}...
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1answer
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Using differentials and given $h=f\circ\vec g$ find $h(1.02,1.99)$

Using differentials find approximately $h(1.02,1.99)$ using that $$h=f\circ\vec g,\quad f(u,v)=3u+v^2,\quad\vec g(1,2)=(3,6),\quad D_{\vec g}(1,2)=\left(\begin{matrix}2&1\\3&5\end{matrix}\...
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1answer
43 views

Given point and normal vector find the derivative of a function

1. State the properties that a $f$ function must have to ensure that its directional derivative $$\begin{matrix}f'\left(\vec A,\check v\right)&=&\nabla f\left(\vec A\right)\cdot\check v&\...
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3answers
58 views

what is the difference between $D(g\circ f)(x)$ and $Dg(f(x))$?

Let $A$ be an open in $\mathbb R^m$. Let $B$ be open in $\mathbb R^n$. Let $f: A \to \mathbb R^n$ and $g: B \to \mathbb R^p$ where $B = f(A)$. If $f$ is differentiable at $a$ and $g$ is ...
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39 views

Wave Equation (Need Clarification Please)

I am supposed to take the Laplace transform of the wave equation that yields a non-homogeneous ordinary differential equation in terms of $\mathcal{L}\{f(x)\}$ and $\mathcal{L}\{u(x,t)\}$ and its $x$-...
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27 views

what is the partial drivetive of vector

I have a problem with finding the derivative of one vector with respect to some of its own term but in a special order. assume that the vector is defined as: $$W=[{w_1,...w_j,...w_n}]^T$$ where $$ ...
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1answer
28 views

Is there a proof for this limit cycle equilibrium

Consider a system of differential equations where both are both continuous partial derivatives. Let's call them $F$ and $G$. Is there a proof suggesting that if there exists a solution that is a limit ...
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77 views

Partial derivative in gradient descent for social recommendations

In paper entitled with "recommendations in signed social networks" by Jiliang Tang, he suggest model for capturing local and global information from signed social networks as follows: $$min \sum_{i=1}...
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Determining surfaces which satisfy the given pde and which circumscribe the given surface

Show that the integral surface of the equation $2y(1+p^2)=pq$ which is circumscribed about the cone $x^2+z^2=y^2$ has equation $z^2=y^2(4y^2+4x+1)$
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3answers
71 views

Calculating $u_t$, $u_x$, and $u_{xx}$ for $u(x, t) = -2 \dfrac{\partial}{\partial{x}}\log(\phi(x,t))$

I am trying to calculate $u_t$, $u_x$, and $u_{xx}$ for $u(x, t) = -2 \dfrac{\partial}{\partial{x}}\log(\phi(x,t))$. I've been trying for hours, but I've become so confused with the chain rule here ...
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2answers
35 views

Doubt about how to find a Lipschitz constant

I have a doubt about a sentence of my Calculus text. Let $f: [t_1, t_2]\times \mathbb{R}^n \to \mathbb{R}^n, (t,y)\to f(t,y)$ such that $|\partial_{y_j} f_i|$ is continuous and bounded for every $...
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1answer
52 views

How to find solution of the following differential equation [closed]

How to find solution of the following differential equation? $\frac{\partial\phi}{\partial t} = a \frac{\partial\phi}{\partial x}$, where $a$ is constant.
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3answers
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Where's my error in this partial derivatives problem? [duplicate]

Let $u(x, y)=x+y$. What is $\displaystyle\frac{\partial u}{\partial x}$ and $\displaystyle\frac{\partial u}{\partial y}$? My answers are $1$ and $1$. Suppose I now told you that $y=x$, so that $u=2x$....
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0answers
39 views

Solving for a stationary point of a multivariate Gaussian parameter (CCA)

I am reading a paper, probabilistic CCA, and am stuck on a particular derivation. Given the following multivariate Gaussian: $$ x \sim N(u, W W^{\top} + P) $$ where $x \in \mathbb{R}^p$. Let $S = W ...
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1answer
47 views

Partial derivatives: Prove $\frac{dx_2}{dx_1}=-\frac{MU_1}{MU_2}$

This is actually an economics question but it involves partial derivatives, so I thought it would be better to ask it here. Let $u(x_1, x_2)$ be a function of 2 variables. Let $\displaystyle MU_1=\...
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2answers
71 views

Why chain rule isn't working here?

Let $(x,y,z)$ and $(x',y',z')$ be two points in space. Let the distance between them be $r$. Also let: $x-x'=\xi$ $y-y'=\eta$ $z-z'=\zeta$ Since $(x,y,z)$ and $(x',y',z')$ are two separate ...
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1answer
44 views

Derivative of a multivariable function at a point [closed]

I'm facing this problem of finding the derivative of a function $f(x,y)= (\sin^2 x \cdot \cos y, xy)$ at the point $(\pi,\pi/2).$ The problem is that I don't know if I should calculate the partial ...
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1answer
63 views

Derivation of derivative of multivariate Gaussian w.r.t. covariance matrix

I'm reading a paper, probabilistic CCA, in which the authors state derivatives without showing derivations. I would like step-by-step derivations to convince myself. Consider a $d$-dimensional ...
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2answers
61 views

Taking derivative of $\exp\bigg(\frac{-x^2}{2 C}\bigg)$ with respect to $x^2$

I am asked to take the derivative: $$\frac{\partial}{\partial x^2} \exp\bigg(\frac{-x^2}{2 C}\bigg)$$ I am told it gives $$\bigg(-\frac{1}{C}+\frac{x^2}{C^2}\bigg)\cdot \exp\bigg(\frac{-x^2}{2 C}\...
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1answer
41 views

Why is the partial derivative of two vectors equal to the vector not taking the derivative

I am not sure how to prove this partial derivative. It seems to me that they are both vectors of integers. So when taking a derivative of the values we get 0 but then the total would be zero overall. ...
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0answers
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Rewriting partial differential of Gaussian PDF

I am given the following partial differential of a Gaussian where the variance is a function of $t$, $\Xi(t)$ as follows: $$\frac{\partial}{\partial t} \frac{1}{\sqrt{2\pi \Xi(t)}}\cdot\exp\bigg[\...
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1answer
49 views

Correct Notation of Mean Value Theorem for Vector-Valued Function

Let ${\bf f}({\bf x})$ be a function ${\bf f}: \mathbb{R}^n \to \mathbb{R}^m$ with continuous derivatives ${\bf H}({\bf x})$. We wish to approximate ${\bf f}({\bf x}_0)$ by ${\bf f}({\bf x})$. It ...
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0answers
41 views

Is the gradient correctly computed?

The function is $\frac{1}{1+\|x_i-c_{yi}\|_2^2}$. Following my computation of the gradient w.r.t. $x_i$. Can somebody please check if it is correct. Let $u = 1+||x_i-c_{y_j}||_2^2$, thus $f = u^{-1}$. ...
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1answer
30 views

Derivative with Respect to a Variable Times Constant? And Then Factoring-Out That Constant as a Fraction?

I have the PDE $\dfrac{\partial{T}}{\partial{t}} = \alpha \dfrac{1}{r} \dfrac{\partial}{\partial{r}}\left( r \dfrac{\partial{T}}{\partial{r}} \right)$ I have done change of variables and now have $r^*...
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0answers
34 views

How can I determine the double derivative of a function $f:R^n \to R$?

A function $f:R^n \to R$ belong to $C^2$. So $Df$ will be a matrix of order "1 by n". So now the range set of function $Df$ will be all row matrix of order "1 by n" while domain set will remain same(...
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0answers
49 views

Can someone help me derive this equation?

I have read this paper Translation-based Recommendations and I have some question about the derivation. I'm not familiar with the derivative of vector. I want to derive $$\frac{\partial (\hat{p}_u,...
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3answers
80 views

Can the non-differentiability of a function $f:R^n \to R$ always be proved by using directional derivative?

$F (x ,y) = |x| + |y|$ when $xy \neq 0$ and $F(x,y) =0 $ elsewhere. How can I prove or disprove this function is differentiable at $(0,0)$? My Try : The directional derivative at $(0,0)$ in ...
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1answer
15 views

Positive directional derivatives on sphere

Let $f:\mathbb{R}^m\rightarrow\mathbb{R}$ be continuous such that all the directional derivatives exist in $\mathbb{R}^m$. If $\frac{\partial f}{\partial u}(u)>0$ for every $u\in S^{m-1}$, then ...