Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

2
votes
0answers
7 views

Bounding the solution to an inhomogeneous heat equation.

I have a PDE question that I am struggling to get started on. Suppose we have a function $u\in \mathcal{C}^2\Big(U\times(0,\infty)\Big)\cap \mathcal{C}\Big(\overline{U}\times[0,\infty)\Big)$ which ...
0
votes
1answer
21 views

Solving a stochastic differential equationn

Does anyone has ideas on how to solve this equation. $$dX_{t} = \left(\sqrt{1+X_{t}^{2}} + \frac{1}{2}X_{t}\right)\,dt + \sqrt{1 + X_{t}^{2}} \,dBt$$ where $Bt$ is a standard Brownian Motion. I have ...
0
votes
0answers
22 views

Chain Rule Second Order PDE

Consider second order PDE: $x\phi_{xx}+2x^2\phi_{xy}-\phi_x=x^3$ My characteristic variables are: $\xi=y-x^2$ and $\eta=y$ $\phi_x = \Phi_\xi.\xi_x+\Phi_\eta.\eta_x=-2x\Phi_\xi$ $\phi_{xx} = -2\...
0
votes
0answers
10 views

Confusion in derivation of “Clairaut's theorem on equality of mixed partials”

Here I came across a derivation of "Clairaut's theorem on equality of mixed partials". In third step, it is written that: With some simple algebraic tidying up, this becomes: In the second ...
1
vote
0answers
18 views

Using the Leibniz integral rule to prove smoothness of the function.

I am working with thee probability density function that depends on non-deterministic ($v$) and random ($x$) parameters (and random parameters depend on non-det parameters): $Pr(v)=\int_{G(v)} dP(v)$,...
3
votes
1answer
30 views

Solve the steady state problem $\frac{\partial u}{\partial t} = \frac{\partial ^2u}{\partial x^2}-e^{rx}$

For the PDE: $\frac{\partial u}{\partial t} = \frac{\partial ^2u}{\partial x^2}-e^{rx}$ , $0<x<L, t>0$ together with the boundary conditions ($r$ and $\alpha$ are constants) $\frac{\...
-1
votes
0answers
21 views

Derivative of composition of functions [on hold]

Suppose $$g:R^p\to R^n$$ is a function of $$ x_1,x_2,....x_p$$ and $$f:R^n \to R^m$$ is a function of $$y_1,y_2,...y_n$$ , if $$h=fog$$ how will we calculate the derivative of h with respect to $$x_1,...
0
votes
1answer
23 views

Ideas to solve the following system of equations

I am trying to solve an exercise with partial derivatives. I have no issue with the derivation part, but once you do that you are supposed to solve a system of equations to find the solution. I ...
1
vote
0answers
66 views

If $u^3+v+w=x+y^2+z^2,u+v^3+w=x^2+y+z^2,u+v+w^3=x^2+y^2+z,$ prove the following

I am stuck with the following problem that says : If$$\begin{align} u^3+v+w&=x+y^2+z^2,\\u+v^3+w&=x^2+y+z^2,\\u+v+w^3&=x^2+y^2+z, \end{align}$$then prove that $$\frac{\partial(u,v,w)...
0
votes
0answers
37 views

Does an integral of a partial derivative make the partial derivative disappear?

I have often seen integrals such as $$A = \int_{t=0}^{t=T} \frac{\partial}{\partial t} \phi(t,x) dt$$ and I'm wondering if the integral cancels the partial derivative when the variable that the $\...
0
votes
1answer
31 views

Derivative of Least Squares with L2 Norm

I'm new to matrix calculus, and I've never really taken derivatives of summations before. Could someone show me how I would get the first order derivative of this? $J(w)=\frac{1}{2}[\sum_{i=1}^{m}(w^...
1
vote
0answers
16 views

Derivation of partial derivative of cost function with respect to weights in backpropagation algorithm

I am studying Machine Learning from Andrew Ng's Machine Learning course on coursera. I am stuck at understanding math behind back propagation. Here is an image of backpropagation algorithm from his ...
0
votes
2answers
18 views

Directional derivative of piecewise defined function?

Let $$f(x,y)=\begin{cases}\frac{x^3+y^3}{x^2-y^2},\ x^2-y^2\neq 0 \\ \ \ \ \ 0 \ \ \ \ ,x^2-y^2=0\end{cases}$$ Then find the directional derivative of $f$ at $(0,0)$ in the direction of ...
-1
votes
0answers
25 views

Finding the equation of a multivariable equation $f(x,y)$ given the equation of $f(x,0)$

If I know that $f(x,0) = g(x)$, How would I find a solution to $f(x,y)$.$$$$ I am also given the equation: $$\frac{f(x,y) - f(x-y,y)}{y} = \frac{d}{dx}g(x); \forall y\in\mathbb{R}$$ My aproach was: $$...
2
votes
1answer
32 views

Using the Fundamental Calculus Theorem for two variables to prove smoothness.

There is a probability density function that depends on non-deterministic ($v$) and random ($x$) parameters: $Pr(v)=\int_{G(v)} dP(v)$, where $G (v)$ is the "goal" region, the probability of getting ...
0
votes
0answers
24 views

Scalar function from $XYZ$ coordinates

As part of the project, I have to calculate the Gaussian, mean, max and min principal curvatures. I understand that I need to use partial derivatives to get them and I understand how to do that but I ...
0
votes
3answers
61 views

How to find the jacobian of the following?

I am stuck with the following problem that says : If $u_r=\frac{x_r}{\sqrt{1-x_1^2-x_2^2-x_3^2 \cdot \cdot \cdot-x_n^2}}$ where $r=1,2,3,\cdot \cdot \cdot ,n$, then prove that the jacobian of $...
0
votes
2answers
23 views

How to find the Jacobian for implicit functions?

I have to find the Jacobian for $$\begin{align} u&= x/ (1-r^2)^{1/2}\\ v&= y/ (1-r^2)^{1/2}\\ w &= z/ ( 1-r^2) \end{align}$$ where $r^2 = x^2 + y^2 + z^2$ but I am not able to solve ...
0
votes
1answer
35 views

I don't understand understand this differential equality

I don't understand the following equality: \begin{equation} x'\frac{\partial x'}{\partial q'}+y'\frac{\partial y'}{\partial q'}+z'\frac{\partial z'}{\partial q'}=\frac{\partial}{\partial q'}(\frac{x'^...
1
vote
1answer
28 views

Can partial derivatives be expressed as a fraction?

Let $a = bc$. Then $b = a/c$. From the first equation, we also have $\frac{\partial a}{\partial c} = b$. Equating, $\frac{\partial a}{\partial c} = b = a/c$, or $\frac{\partial a}{\partial c} = a/c$, ...
2
votes
2answers
42 views

Computing this limit: $ \lim_{y\to0} \frac{f(x,y) - f(x-y,y)}{y} = g(x)$

If $f(x,y) \in \mathbb{R^2}$ and $g(x) \in \mathbb{R}$. Assuming $\frac{f(x,y) - f(x-y,y)}{y} = g(x); \forall y \in \mathbb{R}$ $$$$ Can we do the following: $$ \lim_{y\to0} \frac{f(x,y) - f(x-y,y)}{y}...
0
votes
1answer
35 views

Derivative with respect to vector of product of two functions of the vector

I am struggling with the following derivative. Let $\pmb{x} \in \mathbb{R}^{n}$ be a vector, $\pmb{y} \in \mathbb{R}^{m}$ another vector that is a function of $\pmb{x}$, and $\pmb{g}$ and $\pmb{h}$ ...
1
vote
1answer
34 views

Derivative of a composite function

There are two functions f(x) and g(x): f(x) and g(x) I need to differentiate: (a) g ∘ f using the chain rule (b) h, where h = g ∘ f I found the partial derivatives of f and g with respect to ...
0
votes
0answers
33 views

Finding a multivariable function using it's antiderivative

If I'm given: $\frac{f(x,y) - f(x-y,y)}{y} = g(x); \forall y \in \mathbb{R}$. Can I do the following steps: $$ \lim_{y\to0} \frac{f(x,y) - f(x-y,y)}{y} = g(x)$$ $$ \lim_{y\to0} \frac{\partial{f(x,y)}}{...
0
votes
2answers
54 views

Find if the function $\frac{(1-2xy)}{(x^2 +y^2)}$ has a max or min value for $(x,y)=/=(0,0)$

Does the function $\frac{1-2xy}{x^2 +y^2}$ have a max or min value for $(x,y)=/=0$? What I've tried so far is to take the the partial derivatives: $$\frac{\partial f}{\partial x} = \frac{2(-x+x^2*y ...
0
votes
0answers
14 views

What is the difference between “Uncertainty” and “Propagated Error”

The formula for uncertainty is: u = sqrt( sum[ (partial.i*di)^2 ] ) but the propagated error is found via the total differential: error = sum[ partial.i*di ] Which method is the correct way to ...
0
votes
1answer
20 views

Changing signs of partial derivative's elements

If $C = \frac{1}{2}(y - a)^2$ where $y$ is a given value, $a = \sigma(z)$, and $z = wx + b$. Then the partial derivative of $C$ with respect to w should be: $\frac{\partial C}{\partial ...
0
votes
1answer
32 views

Prove the discontinuity of $f$ at origin.

Prove the discontinuity of $f $ at origin $ f(x,y)=\begin{cases}\frac{x^2-xy}{x+y},&\text{if $(x,y)\neq(0,0)$} \\ 0,&(x,y) \>\ = (0,0)\end{cases}$ . Also show that $f $ has partial ...
1
vote
0answers
53 views

Deriving the error in activation nodes in back propagation algorithm

I am trying to understand back propagation algorithm from Andrew Ng's Machine learning course. Here is a pitcure of the slide on which I am stuck. I know that error in a function $f(x)$ is calculated ...
4
votes
0answers
65 views

Derivation of 2D Korteweg-de-Vries equation

Coming from engineering rather than mathematics, I am recently dealing with non-linear partial differential equations e.g. like the well known Korteweg-de-Vries equation: $$u_{t} + uu_x + u_{xxx} = 0$$...
0
votes
0answers
38 views

Finite difference formula of Third order mixed partial derivative

I know the Finite difference formula of Second-order mixed partial derivative which is: I'm looking for this formula of the third-order mixed partial derivative (i.e. $\frac{\partial^3f(x,y,z)}{\...
3
votes
1answer
41 views

Simplifying the Derivative of a European Call Option

I know the title suggests finance, but I'm stuck on the mathematics of this. I need to take the following derivative: $$ -\frac{\delta C(X)}{\delta X}=-\frac{\delta}{\delta X} \Big[Se^{-dT}N(d_1)-Xe^...
0
votes
0answers
30 views

Partial derivative of a definite integral

I'm creating this algorithm (kinda machine learning) and i have a function which is defined like this: $$ V(x, y) = \int_a^b g(x,y,t) dt $$ (a and b are some constants) And then i want to calculate ...
1
vote
0answers
21 views

Invariance of Solution of Laplace equation on $\mathcal{B}_{1}(0)$ under funny composition

Suppose $v:\mathbb{R}^{2}\rightarrow\mathbb{R}$ solves $\Delta v=0$ on $\mathcal{B}_{1}(0)$. Let $\theta\in(0,2\pi)$ and define $M:\mathcal{B}_{1}(0)\rightarrow\mathcal{B}_{1}(0)$ by $M(x,y)=(x\cos\...
0
votes
0answers
15 views

Second derivate: coordinate free version is equivalent to normal version.

Let $f: A \to B$ be a differentiable function. I know that $Df: A \to Hom(A,B)$ and that $(D^2f): A \to Hom(A,Hom(A,B)) \cong L(A,A; B)$. Furthermore, my teacher proved in class that if $f \in C^2$ ...
4
votes
1answer
43 views

Directional derivative confusion - why does independently evaluating partial changes, then adding them, work?

I apologize for both my crude math grammar, and what is probably an obvious question - I am a novice. I am confused as to why, when taking the directional derivative, the gradient is evaluated by ...
0
votes
0answers
25 views

Arithmetics of '$\wedge$' and '$d$' operators

I don't find arithemtic rules of the operators $\wedge$ and $d$. For example, why does this equality hold? $$ \\ (u^2\cos^2v+u^2\sin^2v)[\cos vdu-u\sin vdv]\wedge [\sin vdu+u\cos vdv] \ \\ +u\cos v[\...
3
votes
0answers
38 views

What is the significance of the half derivative of $x^n$?

I have been researching about half derivatives, and the simplest version of a half derivative I found was $x^n$. However, this half derivative only works when n is a natural number. I feel like it is ...
-1
votes
1answer
29 views

Applying chain rule to $f(x,y) = -h(x,-y)$

Could someone check my work? Thanks! $$-f_x(x,y) = \frac{\partial h(x,-y)}{\partial x}\frac{\partial x}{\partial x} + \frac{\partial h(x,-y)}{\partial (-y)}\frac{\partial (-y)}{\partial x} = h_x(x,-y)...
1
vote
1answer
21 views

Derivative of scalar field and vector field at $(1,2)$ [closed]

Find the derivative of the function at the point $( 1,2 )$: (a) $f ( x , y ) = e ^ { x y }$ (b) $f ( x , y ) = \left( x^2 + y^2, x y \right)$ Are both derivatives matrices?
1
vote
0answers
31 views

Product rule involving partial derivatives

Given: $${\partial \over \partial x} = \cos\phi{\partial\over \partial \rho} -{\sin\phi\over \rho}{\partial\over\partial \phi}$$ I was told to differentiate to get: $${\partial^2\over\partial x^2}=\...
1
vote
1answer
19 views

Euler Lagrange and Geodesics

I’m trying to use the Euler Lagrange equations to derive the geodesic equations. I’ve assumed a lagrangian: $$ L = {1\over 2} g_{ij}\dot x^i \dot x^j $$ So one of the terms of the equation requires: ...
2
votes
0answers
24 views

Solving simple PDE using partial integral

I have a question regarding these notes for solving simple PDE. Here is the image: After finding $P_1, Q_1, R_1$, they have not made it exact differential, as I was taught, but have partially ...
3
votes
2answers
45 views

Non-existence of the potential function

I am wondering why this theorem is not true when $(f,g)$ are defined on a more general open set $U$ which is not necessarily the entire plane or some disc. What is an example of a vector field $$F(x,...
3
votes
0answers
90 views

Converting between Solution forms using Green's Functions in Linear Differential Equation

EDIT: Bounty is over tomorrow so I tried to clean up the question a bit, and put the additional work below as optional to read. I summarized the current results and the solution form I am trying to ...
1
vote
1answer
43 views

Partial derivative of $ f(f(x,-x) , f(x,x)) $.

$ f: \mathbb{R^2} \to \mathbb{R} $ is a function of $C^1$. and $g: \mathbb{R} \to \mathbb{R}$ defined as : $$ g(x) = f(f(x,-x) , f(x,x))$$ Compute $g'(x)$. Suppose $f(0,0) = 0$. ...
0
votes
0answers
28 views

Chain rule questions on partial derivatives

This is a problem in Protter and Morrey's A First Course in Real Analysis. Suppose that $F(x,y,z) = 0$ is such that the functions $z = f(x,y)$. $x = g(y,z)$, and $y = h(z,x)$ all exist by the ...
0
votes
0answers
26 views

Partial Derivative Being Treated As Full Derivative

In this Khan Academy video (https://youtu.be/YT6XwkcPcsw?t=138), Sal takes partial derivatives of several dependent variables. When taking the full derivative of $Q(x,\ y,\ z(x,\ y))$ with respect to ...
1
vote
0answers
18 views

Partial derivatives and normal derivative combined in the chain rule

I have come across the following in some lecture notes and do not understand the interchange of partial derivatives and normal derivatives, by which I mean $\partial$ and $d$, respectively. I am not ...
0
votes
0answers
16 views

Determining partial derivatives after coordinate transformation

I'm currently trying to figure out the partial deriviatives of a function, after there is a space transformation. $$ f(r_1, r_2, w) \rightarrow f(x, y, w)(2\sqrt{xy})^{-1} $$ Where $$ r_1 = x^2 \\ \\...