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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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Confusion in Partial Derivation of an Equation containing Quaternion

I found a way to rotate a 3D vector using a given unit quaternion. Thanks to this answer. Now, let's say I want to rotate a gravity vector: $\overrightarrow{g} = \begin{bmatrix} g_x\\ g_y\\ g_z\\ \end{...
Milan's user avatar
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-3 votes
1 answer
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Let $\frac{\partial}{\partial(\epsilon\,a)}$ [closed]

I have a basic calculus question: Let $\frac{\partial}{\partial(\epsilon\,a)}$; $\epsilon$ is a small constant. How can I rewrite the partial derivative: Is it the same as $\frac{1}{\epsilon}\frac{\...
s28's user avatar
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Is it possible to let $\frac{\partial f(x,y)}{\partial (-y)}=\frac{\partial f(x,a-y)}{\partial (a-y)}$ hold, where $a$ is a constant? [closed]

Let $f(x,a-y)$ be a function that is continuously differentiable in $y$, and $a$ be a constant. Under what conditions does the following equality hold? \begin{equation} \frac{\partial f(x,y)}{\partial ...
Jeff Hsieh's user avatar
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Chain rule when mixed partial multiplied by first order partial

Given $$ r = \frac{{\partial s}}{{\partial t}} $$ If we take the partial derivative of $r$ with respect to $u$ $\frac{{\partial r}}{{\partial u}}$ and multiply it by $\frac{{\partial u}}{{\partial v}}$...
MikeM's user avatar
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Is this a valid notation for a directional derivative?

In Siegel (2005): Fields I found the following strange notation on page 170 $$ \frac{\partial f(\phi_i)}{\partial \phi_j} = \lim_{\epsilon\to0}\frac{f(\phi_i + \epsilon \delta_{ij})-f(\phi_i)}{\...
asmaier's user avatar
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1 answer
67 views

Convert $\partial_{\beta}\partial_{\gamma}(\epsilon_{\alpha\gamma\nu}p_{\nu}p_{\beta})$ from index notation to vector notation

I have the expression $\partial_{\beta}\partial_{\gamma}(\epsilon_{\alpha\gamma\nu}p_{\nu}p_{\beta})$ written using Einstein summation convention with $\alpha$ the only free index, p is a vector in ...
Archie Brew's user avatar
2 votes
1 answer
66 views

Confusion on Notations of Partial Derivatives on Manifolds

I'm confused with the notations of partial derivative on manifolds in Tu's An Introduction to Manifolds.. Just to make clear the notations I'm using, what I've known and which part I'm confusing, ...
TheHan6edMan's user avatar
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36 views

composite derivative and variable change

I'm asking for help with this calculation that I've been racking my brain over for two days! We have a variable change: $$p=e^{-\gamma t}\hat p$$ $$dp = d \hat p e^{-\gamma t}$$ Now the pdf becomes $$\...
Ged's user avatar
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Changing coordinates of $2$nd order partial operators

Let's work in $\mathbf R^n$. If we want to change coordinates $\mathbf x\to\mathbf r$, with them related like $$ \mathbf x=\mathbf x(\mathbf r) $$ Then, the second order generic partial operator in ...
Conreu's user avatar
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2 votes
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Derivative of the determinant of a matrix with respect to the components

Given a square matrix $A$ $$ A = \begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots &\vdots\\ a_{n1} & \cdots & a_{nn} \end{bmatrix} $$ what is the ...
duc4rm3's user avatar
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Gradients of vertices on a grid

Say we have an irregular 2D grid, it can be viewed as a group of triangles composed, and we know the value of function v(x,y) on the plane at each vertex on the grid (The picture is just for ...
user900476's user avatar
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40 views

A question about the definition of derivative in different coordinate systems

The definition of the derivative goes like this: If $x$ is an interior point of a set $E \subseteq {\Bbb R}^n$, then a function $f: {\Bbb R}^n \rightarrow {\Bbb R}^m$ is said to be differentiable at $...
WhyNót's user avatar
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2 answers
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Implicit differentiation choice

I was reading Calculus early transcendentals by Howard Anton, in which I encountered an example as follows, Find the slope of tangents of a sphere $x^2+y^2+z^2=1$ in the direction of $y$ at points $(2/...
Kaustubh Limaye's user avatar
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Can the partial derivative of a multivariable function be such that it does not depend on the other variable? [duplicate]

I wanted to know if there is a multivariable function out there which when partially differentiated with respect to x returns a function only in x.(Not dependent on y(the other variable)). For example-...
Aditya Saraswat's user avatar
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Help with multivariate function derivative algebra... Middle terms don't match as expected

Intro I want to compute $\frac{\partial^2 f}{\partial m^2}$ where we define $ f(m,\sigma) = g(m, I(m, t)) $ I asked about this in a prior question, however I'm realizing that I don't actually ...
financial_physician's user avatar
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Derivation of equation from Euler's equation for a fluid

I'm having trouble with the formula for particle velocity of a standing wave in a tube, starting from conservation of mass and Euler's equation for a fluid. I've checked a few books and unfortunately ...
korokame's user avatar
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Information coefficient as loss function of XGBoost

$$ IC = \frac{\frac{1}{n}\hat{y}^Ty-\mathrm{E}\left[ \hat{y} \right] \mathrm{E}\left[ y \right]}{\sigma \left[ \hat{y} \right] \sigma \left[ y \right]} $$ XGBoost requires a gradient and a Hessian of ...
atlantic0cean's user avatar
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Interpreting the partial derivative of a low-pass filter with respect to its parameter

Looking at the Laplace transform of a single pole low pass filter and its partial derivative with respect to its 'time constant' $\tau$: $$ \begin{align} G(s, \tau)=\frac{1}{1+s\tau},&& \frac{\...
MartinC's user avatar
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How to prove this function is differentiabile at this point?

Consider the function $f(x, y) = \begin{cases} e^{x} & x \leq y \\ e^{y} & x > y \end{cases}$. I'm asked to prove through boundings and majorisations that/if $f$ is continuous at $(1, 1)$ ...
Heidegger's user avatar
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How do I find maxima at a point when 2nd partial derivative test fails? [duplicate]

$\mathbb{f}(x,y) = 2(x-y)^2-x^4-y^4$ I found three points as candidates for the maxima $(\sqrt{2}, -\sqrt{2}), (-\sqrt{2}, \sqrt{2}), (0,0)$ At these three points, I evaluated $D = \mathbb{f}_{xx}\...
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Find partial derivative $\frac{\partial (xyf(xy))}{\partial y}$ given $y=g(x)$.

Given $y=g(x)$, Find partial derivative $\frac{\partial (xyf(xy))}{\partial y}$. There are two ways of doing this. Use chain rule, we get the derivative$=xf(xy)+xyf_y(xy)=xf(xy)+x^2yf'(xy)$. ...
High GPA's user avatar
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How is the derivative with respect to $x$, $y$, $z$ found of $F(x,y,z) = x^2 y + x^2 x^3 + y^4$? [closed]

How is the derivative with respect to $x$,$y$,$z$ found of $F(x,y,z) = x^2 y + x^2 x^3 + y^4?$
Ariana Riobueno Marquez's user avatar
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1 answer
20 views

Jacobian chain rule equation: proving an identity.

I have the following problem which I am unable to solve expanding out. It would be kind if someone can provide me some pointers as to how to solve the problem. if $$ x = f(u,v,w), y=g(u,v,w), z=h(u,v,...
user1612986's user avatar
2 votes
1 answer
65 views

Schwartz theorem

Can i relax the conditions on f of the Schwartz theorem in real analysis? That is: if $f: \Omega \to R^2$, s.t $f \in C^2(\Omega)$, then i can change the order of derivatives and the result is the ...
Lucio Rosi's user avatar
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29 views

how to solve the partial derivative of layernorm in neural network: $\frac{\partial y}{\partial x}$ or $\frac{\partial loss}{\partial x}$

give eq as: $$ y (loss) = \frac{x-\mathrm{E}[x]}{\sqrt{\operatorname{Var}[x]+\epsilon}} * \gamma+\beta = norm * \gamma + \beta $$ $\gamma$ and $\beta$ are indepandant of x. norm defined as: $$ norm = \...
melon's user avatar
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Calculus: Partial Derivative Equivalencies

Is the following expression true? $$\frac{\frac{\partial\text{u}}{\partial\text{x}}}{\partial\text{y}}=\frac{\partial\text{u}}{\partial\text{x}\partial\text{y}}=\frac{\partial\text{u}}{\partial\text{y}...
s28's user avatar
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5 votes
1 answer
64 views

Question About Comparative Statics and Optimization

Let me try to describe my question first. By solving the following maximization problem $$ \max_{t_P} U_{B_i}(t_P) = (1-t_P)y_{B_i} + V_{B_i}(g), $$ where $V_{B_i}$ is a function of $g$ and $g=t_P\...
Beerus's user avatar
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1 vote
0 answers
35 views

Study the variation of a function according to changes of a constrained variable

I am working on the maths behind constant product market makers (CPMMs). The main underlying equation is $R_XR_Y = K$ where $R_X$ and $R_Y$ are the reserves of two assets, and $K$ is a constant. When ...
Riccardo Perego's user avatar
1 vote
1 answer
80 views

How to take the partial derivative of a functional.

Given a function: $$F(3x, 5xy) = 4(3x) + 5xy$$ And we want to know the answer of $\frac{\partial F}{\partial x}$: $$\frac{\partial F}{\partial x} = \frac{\partial 3x}{\partial x}\cdot\frac{\partial F}...
zizaaooo's user avatar
3 votes
1 answer
48 views

Why isn't continuity necessary for existance of partial derivatives?

Consider the graph of a function. $$f(x,y) =\begin{cases} \frac{xy}{x^2+2y^2}, & \text{(x,y) $\neq$ (0,0)} \\ 0, & \text{elsewhere} \end{cases}$$ It is discontinuous at $(0,0)$. As on path $y ...
androidDeweleper's user avatar
0 votes
2 answers
63 views

Finding the derivative of trace $AXBXC^T$ with Respect to $X$

If A,B,C,X are matrices, find: $$\frac{\partial \text{tr}[AXBXC^T]}{\partial X}.$$ Here's my initial approach: $$\partial \text{d} \text{tr}[AXBXC^T] = \text{tr}[\text{d}(AXB)XC^T] + \text{tr}[AXB \...
Fernand's user avatar
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0 votes
0 answers
19 views

Multivariate quotient rule for partial derivatives of dot product between normalized (unit) vector and another vector with respect to first vector

I would like to confirm that the following is a correct application of a multivariate quotient rule as explained in the answer to Quotient rule extendable to functions of vectors? For any vectors $x=[...
leka0024's user avatar
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0 answers
20 views

Chain rule in differential of Gauss map

This is probably more of a calculus question than a geometry question. Let $N: S \rightarrow \mathbb{S}^2$ be the Gauss map. And let $\varphi : \mathbb{R}^2 \supseteq U \rightarrow S$ a ...
F13's user avatar
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0 votes
1 answer
58 views

Newton-Raphson algorithm proof: derivating a matrix

A particularly useful algorithm when you want to find the zero of a function is the Newton-Raphson method. For simplicity, we begin by examining the simplest case. Given a function and his root $x^\...
user3204810's user avatar
0 votes
3 answers
70 views

How to understand the partial derivative of a total derivative?

I am going through a thermodynamics course and there is a section with a derivation that I don't quite understand from a mathematical standpoint. Ignoring chemical potential for now, the "natural ...
xoux's user avatar
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Partial derivatives with respect to position of rate of change of distance (Euclidean norm) between two time-varying positions

I'm interested in the partial derivatives $$\frac{\partial d}{\partial p_x}, \frac{\partial d}{\partial p_y}, \frac{\partial d}{\partial p_z}\quad\text{,}\quad\frac{\partial\dot{d}}{\partial p_x}, \...
leka0024's user avatar
1 vote
1 answer
48 views

How to calculate the partial derivates in the origin $(0,0)$

Study the continuity, derivability and differentiability in the origin of the following function: $$ f(x,y)=\begin{cases}\dfrac{xy^2}{x^2+y^2+|xy|}&\quad\text{when}\:(x,y)\neq(0,0)\\0&\quad\...
Sebastiano's user avatar
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0 votes
1 answer
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Question on partials

Let $w = f(x, y, t)$ and $x$ and $y$ depending on $t$. Suppose that at some point $(x, y)$ and at some time $t$, the partial derivatives $f_x$, $f_y$, and $f_t$ are equal to $2$, $-3$, and $5$, ...
secretrevaler's user avatar
3 votes
1 answer
35 views

Partial derivatives and functions

If I have $y=x^2$ and I have a function of $x$ and $y$ i.e. $f(x,y)=x+y$, then why is it that the partial derivative of this function with respect to $x$ is 1 whereas the partial derivative of $g(x)=x+...
secretrevaler's user avatar
1 vote
1 answer
43 views

Prove that in the canonical basis $\mathrm{grad}\,f = \sum_{j=1}^{n} \frac{\partial f}{\partial x_j}\,e_i$.

Reading Do Carmo I found this exercise: Given a differentiable function $f:\mathbb{R}^n \to \mathbb{R}$ the vector field (gradient) is defined by $$ \langle \mathrm{grad}\,f(p) \rangle =df_p(u) $$ ...
Daniel R.S's user avatar
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99 views

Gradient of softmax function with the inner product argument

Suppose that $x_i$ and $y_j$ are vectors in $\mathbb{R}^d$, where $i,j\in\{1,2,\dots,N\}$. Let the loss function be defined as $$\mathcal{L} = -\frac{1}{N}\ln\left(\frac{\exp({\frac{x_i.y_i}{\tau\...
S.H.W's user avatar
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0 votes
0 answers
37 views

A clever way of calculating sum of partial derivatives

Given the function in 4 variables $f(x, y, z, t) = \frac{x - y}{x - t} + \frac{t - x}{y - z}$ prove that $f'_x + f'_y + f'_z + f'_t = 0$. Of course one can prove this by mechanically carrying out all ...
Moxy's user avatar
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0 votes
2 answers
80 views

Partial derivatives of this function: it does exist but it doesn't.

I don't understand why I'm getting two different results about the existence of the partial derivative wrt $x$ of this function : $$f(x, y) = \begin{cases} \frac{x^2+2y}{x^2+y^2} & (x, y) \neq (0, ...
Heidegger's user avatar
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1 vote
1 answer
23 views

What happen if CDF or survival of bivariate distribution is differentiated with one component only?

Hello brilliant people, I have some questions regarding joint probability distribution. Let say I have a joint density $f(x,t)$. A joint CDF is computed as follows, $$ F(x,t)=P(X<x,T<t)=\int_{-\...
AHMAD FAIZ BIN MOHD AZHAR MSC2's user avatar
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0 answers
42 views

Question about partial derivative of logarithm

Let $\Omega$ be an open set and $f \in H(\Omega)$. I want to compute $\frac{\partial \log(f\overline{f})}{\partial \overline{z}}$. What I've tried: $\frac{\partial \log(f\overline{f})}{\partial \...
MathLearner's user avatar
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1 vote
2 answers
102 views

Why does the partial derivative of $y'$ with respect to $y$ vanish? [closed]

I'm following Weinstock's Calculus of Variations. On page 25 it says: We have in this case $$f=\frac{d g}{d x}=\frac{\partial g}{\partial x}+\frac{\partial g}{\partial y}y',$$ so that the Euler-...
sensorer's user avatar
0 votes
1 answer
43 views

Partial derivatives for functions with one complex variable

So far, suppose that we defined partial derivatives for functions $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$. In Cauchy-Riemann equations, for $f: \mathbb{C} \rightarrow \mathbb{C}$, we first view $f(...
Gunt Ryumet's user avatar
2 votes
1 answer
61 views

Multivariable chain rule for function with one negative component

I have a (smooth) function $\psi:\mathbb{R}^2 \to \mathbb{R}\: : \: (y_1,y_2) \mapsto \psi(y_1,y_2)$. Now I want to calculate the partialderivative of $\psi(-y_1,y_2)$ with respect to $y_1$. I thought ...
want2know's user avatar
0 votes
1 answer
30 views

Calculate mixed second partial derivative

Given a function $f: \mathbb{R}^2 \to \mathbb{R}$ with continuous partial derivatives. It is given in addition that: [ f'x (3,9) = f'y (3,9) = f''{xx} (3,9) = f''{yy} (3,9) = 1 ] Define $g(x, y) = f(x^...
Roei's user avatar
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0 answers
26 views

What if the dot product of gradient vector of two curves is zero at a point (x,y)?

If say the grad(f)•grad(g) at point (x,y) on both the curves is zero. Is it sufficient to show that the curves intersect orthogonally?
Aniket Sinha's user avatar

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