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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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How to find the function $Y(K,L)$?

$\alpha$ is just a constant and it's given that $$\frac{\partial Y(K,L)}{\partial K}=\alpha\frac{Y}{K}$$ $$\frac{\partial Y(K,L)}{\partial L}=(1-\alpha)\frac{Y}{L}$$ Doing some integration $$\ln(Y)=...
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From $dxdy$ to $\rho d\rho d\phi$. Where am I doing wrong?

A small area element in the xy plane reads $da=dxdy$. In plane polar coordinates, it reads $da=\rho d\rho d\phi$. We also know, $$x=\rho\cos\phi,~ y=\rho\sin\phi.$$ So using partial derivative formula,...
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Laplace Equation for gravitational potential [on hold]

While deriving the Laplace Equation for gravitational potential by the method described below, can someone please explain as to how the last term of the equation (3.6.9) has been equalled to zero? ...
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$\frac{-\partial u}{\partial t}\frac{\partial^2 u}{\partial x^2}= \frac{\partial^2 u}{\partial t \partial x}\frac{\partial u}{\partial x}$

In Proving energy conservation for wave equation , it is shown how to prove that solutions to the wave equation conserve their energy. The answer includes the relation: \begin{equation} \frac{-\...
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How to derive this complex differentiation?

In the finite-deformation theory, the elastic Cauchy-Green strain $\mathbf{E}_e$ is defined as $\mathbf{E}_e=\frac{1}{2}(\mathbf{F}_e^T \mathbf{F}_e-\mathbf{I})$, where the superscript $T$ denotes ...
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Reference for the multivariate Leibniz rule of many factors

I'm looking for a reference (a book/article) with a formula to $$ \frac{ \partial ^ k }{ \partial x_1^{k_1} ... \partial x_n^{k_n} } f_1(x) ... f_m(x) , $$ where $k=k_1+...+k_n$, $x=(...
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Using partial derivatives to find normal vector

So, I can not find out what I'm doing wrong with this question, even if my life depended on it. I know instruction for doing it, but I can't seem to figure out what I'm doing wrong. Because I refuse ...
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Partial Derivative Disambiguation

There are at least two substantially different meanings to $\frac{\partial}{\partial x}f(x,\ y,\ z(x))$. The $\partial x$ could mean "with respect to $x$ the independent variable," or it could mean "...
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Changing to slanted coordinates

In order to convert a partial derivative of an arbitrary function of two variables from Cartesian coordinates to polar coordinates, we can simply employ the chain rule as follows: $$ \frac{\partial f(...
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Computing partial derivatives of $f(a,b)= \int_{0}^{1}(ax+b+\frac{1}{1+x^2})^{2}dx$ using chain rule.

Let $f(a,b)= \int_{0}^{1}(ax+b+\frac{1}{1+x^2})^{2}dx$. I want to compute $\frac{\partial{f(a,b)}}{\partial{a}}$ and $\frac{\partial{f}(a,b)}{\partial{b}}$. I was told in the text that $$\frac{\...
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Euler Lagrange equations, chain rule troubles

When considering the first integral the chain rule is used on $$F(y,y’,x)$$ When we do this why do we not consider it as $$F(y(x))$$ As y’ is a function of y But instead as y’ being a separate ...
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How to evaluate this partial derivative in terms of polar coordinates

How to evaluate this partial derivative in terms of polar coordinates? How to solve this question?
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Question related to sigmoid function

I have been going through the Deep Learning course from http://neuralnetworksanddeeplearning.com/chap1.html I have come across this approximation in the function $w_jx+b$ $\Delta output \approx \...
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Prove Jacobian of $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ with 3 conditionals over $\mathbb{R}^{2}$ is $I_{2 \times 2}$.

If $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ is given by: $f(x,y)= \begin{cases} (x,y-x^{2}) & if & x^{2} \leq y \\ (x,\frac{y^{2}-x^{2}}{x^{2}}) & if & 0 \leq y \leq x^{...
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An integral inequality on a disc about twice derivatives.

Denote $f:\mathbb{R^2}\rightarrow\mathbb{R}$ has a continuous twice derivative on the disc $D=\{(x,y):x^2+y^2\leq1\}$, which satisfy $f(\partial D)=\{0\} $, where $\partial D=\{(x,y):x^2+y^2=1\}$, ...
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Derivative of integral for non-negative part functions

Suppose we have $N$ random variables $u_1,u_2,...,u_N$, which are i.i.d with PDF $f(\cdot)$. Then how to compute the partial derivative of $g(x,y)$ with respect to $x$ and $y$? That is, $\frac{\...
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Particle moving clockwise along the unit circle centered at the origin of the xy-plane [on hold]

A particle moves clockwise along the unit circle centered at the origin of the xy-plane. Find the direction of $\nabla\times\mathbf v$. My attempt: I found that $\displaystyle\frac{\partial v_x}{\...
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Derivative of $C = D_KL(p || q)$ w.r.t $q_{ij}$ where $q_{ij} = \frac{\exp(z_{ij})}{\sum_{k=1}^N\sum_{l=1}^N \exp(z_{kl})}$

For an exercise, I need to compute $$\frac{\partial C}{\partial q_{ij}}$$ where $$C = D_{KL}(p || q) = \sum_{i=1}^N \sum_{j=1}^N p_{ij} \log \left (\frac{p_{ij}}{q_{ij}} \right)$$ and $q_{ij} = \...
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When do derivatives cancel inside integrals when working with tensors?

While doing a problem recently I realised I'm not clear about when derivatives inside integrals will cancel when working with tensors. For example, I have come across integrals such as: $\int \...
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1answer
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Prove $\exists\theta\in(0,1)$ s.t. $\Delta f=\frac{\partial f}{\partial x}\Delta x+\frac{\partial f}{\partial y}\Delta y$

Let $f(x,y)\in C^1$ in $\mathbb{R^2}$ and let $(x_0+\Delta x,y_0+\Delta y)$ and $(x_0,y_0)$ be points in $\mathbb{R^2}$. Prove that $\exists\theta\in(0,1)$ such that: $$f(x_0+\Delta x,y_0+\Delta y)-...
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Is it true $\int_0^k \frac{\partial f(x,y)} {\partial x}\,dx = f(k,y) + \textrm{a function of }y$?

I saw it written somewhere that $$\int_0^k \frac{\partial f(x,y)} {\partial x}\,dx =f(k,y) + \textrm{a function of }y$$ This seems feasible, but I haven't seen the integral of a partial derivative ...
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1answer
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Partial derivative of $f(u,v)$

Let $f(u,v) = c$ where $u(x,y) , v(x,y)$ are functions and $c$ is constant. Can we conclude $\frac{\partial f}{\partial v} = \frac{\partial f}{\partial u} = 0$ ? It really sounds confusing to me but I'...
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Total and Partial Derivatives With Multiple Implicit Independent Variables [closed]

Given a function of function(s) of multiple independent variables, i.e., $f(x(s,\ t))$ or $f(x(s,\ t),\ y(s,\ t))$, does $f$ have a total derivative? Does $f$ have partial derivatives with respect to ...
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1answer
39 views

A differentiable multivariable function

I know the chain rule for the multivariable functions works when the partial derivatives are continous but if the function is just differentiable does the chain rule work ? I mean if $z = f(x,y)$ is ...
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34 views

Derivatives for box integral of a bivariate normal distribution

I'm having quite a bit of trouble trying to understand how to to calculate partial derivatives of a specific function. Suppose I have the standard bivariate normal density function: $$f(x,y,\rho)=\...
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1answer
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Calculating the partial derivative

I know the definition of the partial derivative but there is another way for calculating the partial derivative by holding one variable constant and taking ordinary derivative . When this method works ...
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Directional derivative $f(x,y,z) = \sin(xyz) $

Using the limit definition, find the direction derivative of $f(x,y,z)=\sin (xyz)$ at the point $P(1,1,0)$ in the direction of $v=⟨2,1,1⟩$. $D_vf(a) := \lim_{t \to 0} \frac{f(a+tv) - f(a)}{t}$ ...
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What is the complete Partial Derivatives approximation formula to calculate the value of a function?

We know the approximation formula using partial derivatives to calculate the value of a function with some variables $x$ and $y$, which can be extended to any number of variables to be \begin{...
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1answer
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Partial Integration wrt a different variable [closed]

I'm interested in how I would integrate $$ \int_0^1 U_{tt}(t,x) U_t(t,x)\ dx \quad \text{ and } \int_0^1 U_{xt}(t,x)U_x(t,x)\ dx $$ Thanks!
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Schwarz's Theorem in n variables

I need the proof of the Schwarz's Theorem in n variables. I did not find it in books and on the web (I found only the proof in R^2).
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1answer
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PDE for heat conduction with loss

Consider the following PDE: $$\frac{\partial }{\partial t}u = \alpha^2 \frac{\partial^2 }{\partial x^2}u -bu, \ \ b>0, 0<x<L$$ The problem asks to "set the time derivative in the PDE to ...
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Solve coupled partial differential equations

I'm trying to simulate lava flow dynamics. May someone help me with solving this pde; The pde and it's solution id given in the image below image link : Que and solution
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Integral of partial derivative: $\int_0^\mu \frac{\partial q} {\partial u}\,du$

An equation (borrowed from statistics QLE) is $$ \frac{\partial q(\mu,y)} {\partial\mu}= \dfrac{y-\mu}{a_i \phi V(\mu)}$$ Apparently this is equivalent to $$q(\mu,y)=\int_0^\mu \frac{y-u}{a_i \phi ...
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Derivative of a Partial derivative in wave equation

Given that: $$y=f(x,t)$$ Prove that: $$\frac{d}{dx} \frac{\partial{y}}{\partial{x}} = \frac{\partial^2{y}}{\partial{x}^2} $$ I have zero experience in PDE's , I just stumbled across this stuff while ...
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How to find intersecting point of following type of functions?

We are given a function,$$f(x,y)=4x^2-xy+4y^2+x^3y+xy^3-4$$ Now to calculate minimum and maximum value of $f(x,y)$, i first calculated $f_x$ and $f_y$ for stationary points. which gave $$f_x=8x-y+3x^...
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Finding points which are not local maximum or minimum

Consider the picture below: This is the levels curves for a function $f(x,y)$ where: Blue line is the partial derivative of $f(x,y)$ with respect to x Red line is the partial derivative of $f(x,y)$ ...
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An expression for computing second order partial derivatives of an implicitely defined function

Let $\Phi(x,y)=0$ be an implicit function s.t. $\Phi:\mathbb{R}^n\times \mathbb{R}^k\rightarrow \mathbb{R}^n$ and $\det\left(\frac{\partial \Phi}{\partial x}(x_0,y_0)\right)\neq 0$. This means that ...
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Hessian of quadratic forms

can you give me a proof that the Hessian matrix of a quadratic form woth associated symmetrix matrix A is equal to 2A? I do not understand all the other proofs I have found.
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partial derivative of definite intergral

Question: Find the partial derivatives, $f_x(x, y)$ and $f_y(x,y)$, of the function $$f(x,y)=\int_y^xcos(3t^2+9t-1)dt$$ My attempt is as follows. Substitution: $u=3t^2+9t-1$ $\frac{du}{dt}=6t+9$ $...
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1answer
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Partial derivatives (calculus)

I am new to calculus and am somewhat confused by the details of partial derivatives in more complex constructions. I was asked to work out the following and was wondering if anybody could give me some ...
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Taking the derivative of a differential equation

My book jumps from $$\frac{\partial f}{\partial x}(x, g(x)) + \frac{\partial f}{\partial y}(x, g(x))g'(x) = 0 $$ to $$\frac{\partial^2 f}{\partial x^2}(x, g(x)) + 2 \cdot \frac{\partial f}{\partial ...
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Partial derivative of a multivariate function of a function

I may be missing something super simple here, but it's Friday and my brain can't handle this. I'm looking at the partial derivative of a multivariate function of the form: $\frac{\partial}{\partial ...
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2answers
70 views

Partial Derivatives of y and y'

In this application of the Euler-Lagrange equation, it is said that there is no $y$ in the function $\sqrt{1 + (y')^2}$. I see that the algorithm in progress treats $y'$ as unusually autonomous, as ...
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log-transforming and finding partial derivatives

I trying to set up an economic model of trade, and i have a question regarding the partial effect of one of my variables that is logged along with my general function $X$. My model of bilateral ...
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About derivative of delta function - chain rule for delta function containing a function

I have a problem that relates to derivative of a delta function. The problem originates from a paper I was reading https://aip.scitation.org/doi/full/10.1063/1.2938860 In the paper, it is said that ...
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Parameters estimation (fit) sensitivity

I have a linear fit estimation of the following type: y=ax+b I would like to test how much a mismatch in a and b affects the accuracy of y? should I use partial derivative or some other method?
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Shock-like phenomena modelled by the 1-D wave equation

I am attempting to determine the behaviour of characteristic lines that arise from my solution to the PDE: $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0$ Assuming that $u(x, 0)$ ...
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1answer
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$u(x,y)=H(x-y)$ is a solution of a partial differential equation

I am trying to prove that $u(x,y)=H(x-y)$ ($H$ denotes the heaviside function) is a solution of the partial differential equation $$\frac{ \partial^2 u}{\partial x^2} -\frac{ \partial^2 u}{\partial y^...
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1answer
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How do I write code for solving partial derivatives numerically?

As stated in the title. I am trying to write a function which evaluates the partial derivative at two points (a,b) for f. However, the output of the partial derivative evaluated at (0,0) is way too ...
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1answer
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change of variables in a differential equation with partial derivatives?

I have a doubt about the change of variables in differential equation. Suppose to have: $\frac{\partial f}{\partial t} + a \frac{\partial f}{\partial x} = b \frac{\partial^2 f}{\partial x^2}$ Now ...