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Questions tagged [chain-rule]

For questions involving the chain rule in analysis.

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28 views

Solving an equation that contains a variable which brings 0=0 issue

So I encountered this question: Given that the relationship between distance (m) and velocity (v) of an object is $$v^2 = 1 - m^3$$ Find the acceleration of the object when $m=1$ By taking the ...
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1answer
21 views

How to use the chain rule for change of variable

I have asked this questions: Change of variables in differential equation? ...but after thinking about it, I am still a little confused of how to rigorously use the chain rule to calculate the ...
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1answer
23 views

Laplacian in elliptical coordinates

I'm trying to calculate the laplacian in elliptical coordinates, just with the chain rule (because I don't know other method for doing this), but I have found difficulties to find the right expression....
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1answer
31 views

Matrix Differentiation of Kronecker Product

I have a question about differentiating an expression which has multiple kronecker products. I have the following objective function I would like to differentiate with respect to $\mathbf{Q}$: \...
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2answers
38 views

Chain rule doubt

I have a doubt of appling the chain rule. I have this $L$ function: $$ L = y\cdot log(\frac{e^{a x+b}}{e^{ax+b} + exp^{cx+d}}) $$ I can rewrite it as: $$ L = y\cdot log(p) $$ where $$ p = \frac{e^{v_{...
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21 views

Rules for inverse functions and partial derivatives

I have that $u(t,x)$ satisfies $\partial u/\partial t + u \cdot \partial u/\partial x = 0$ I need to show that if $x = x(t)$, then $dx/dt = u(t,x)$ So far I have $u = -\partial u/\partial t \...
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2answers
28 views

$\frac{\partial{z}}{\partial{y}}$ if $z$ is given implicitly

Suppose $z$ is given implicitly as: $$e^z-x^2y-y^2z = 0$$ Find $$\frac{\partial{z}}{\partial{y}}$$. I let $F(x,y) = e^z-x^2y-y^2z$. Then, $$\frac{\partial{F}}{\partial{y}} = -x^2-2yz$$ $$\frac{\...
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2answers
42 views

Prove the definition of the arcsin(s).

I am given $\arcsin: S \rightarrow (-\pi/2,\pi/2) $ is the inverse function of sin(t) (restricted to [$-\pi/2,\pi/2$]). I'm trying to prove that $\arcsin(s)$= $\int_{0}^{s}1/\sqrt{1-x^2}$ . My ...
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1answer
49 views

Chain rule when applying L'Hopital's rule

I have a very basic question regarding derivation function: $$f(\omega(t)) = \frac{2 +x(t)\cdot \frac{d\omega(t)}{dt}}{\omega(t)} $$ when I check for $$= \lim_{\omega(t)\to\ 0}\frac{2 +x(t)\cdot\...
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0answers
29 views

Is there a version of the chain rule that applies to hessian matrices?

Suppose I have a scalar $J(n)$ and two vectors, $\mathbf{w}(n)$ and $\mathbf{x}(n)$. Now, suppose that $J(n)$ is a fairly straightforward function of $\mathbf{w}(n)$, and $\mathbf{w}(n)$ is actually a ...
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2answers
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Chain rule - Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y} $ if $z=pq+qw$

Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y} $ if $z=pq+qw$ $p=2x-y$, $q=x-2y$ and $w=-2x-2y$ Is $\frac{\partial z}{\partial x}$ equals to: $\frac{\partial z}{\...
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0answers
10 views

Laplace to gradient chain rule

how do I do the chain rule for f(-$\Delta$)gh if f,g,h are functions and I want it written in terms of gradients. I know f(-$\Delta$)g= $\nabla$f x $\nabla$g.
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21 views

Chain rule in the context of backpropagation in Recurrent Neural Networks - Understanding a derivation

I am studying on my own about RNNs and particularly backpropagation. I have found in the web a slide presentation explaining backpropagation step by step but I am stuck in a particular slide I can ...
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2answers
53 views

Is integration by substitution always a reverse of the chain rule?

To integrate $\int x^3\sin(x^2+1)dx$, I took the following approach: \begin{align*} \begin{split} \int x^3\sin(x^2+1)dx&=\int x^3\sin(u)\cdot\frac{1}{2x}du\\ &=\frac{1}{2}\int x^2\sin(u)du\\ &...
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3answers
47 views

Derivatives of functions composition: $\lim_{x\rightarrow 8}\frac{\root{3}\of{x} - 2}{\root{3}\of{3x+3}-3}$

I have to calculate the folowing: $$\lim_{x\rightarrow 8}\frac{\root{3}\of{x} - 2}{\root{3}\of{3x+3}-3}$$ I am not allowed to used anything else than the definition of the derivative of a function $...
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1answer
31 views

Derivation $f(tx) = t^{\mu}f(x) \text{ } \forall x \in \mathbb{R}^n \text{\ {0}} \text{ } \forall t \in \mathbb{R}^+$

Let $\mu \in \mathbb{R}$ be a real number and $f:\mathbb{R}^n$\ {$0$} $\to \mathbb{R}$ a function that is positive homegenous with degree $\mu$, which means: $$f(tx) = t^{\mu}f(x) \text{ } \forall x ...
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1answer
22 views

chain rule of a second derivative

Suppose I have the following function where $$z=\omega(\zeta)=\frac{1}{\zeta}$$ and also, $$\phi(\zeta) = \zeta^{-1}+2\zeta$$ By using chain rule, I can get the first-derivative of $\phi(z)$. Notice ...
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2answers
79 views

Problem with derivative of $x^{x^x}$

I was recently watching blackpenredpen’s video (found here: https://m.youtube.com/watch?v=UJ3Ahpcvmf8) where he found the derivative of the the function $y = x^{x^x}$. Before watching the video, I ...
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0answers
28 views

Applying Chain Rule to Dimensionless Transformation

Hello I am trying to show that the equation $\frac{dN}{dt} = rN(1 - \frac{N(t - \tau)}{K})$ can be rewritten in a dimensionless form as $\frac{dy}{dx} = \lambda y(1 - y(x - 1))$ using the ...
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1answer
37 views

How to apply the chain rule in vector calculus with inverses and with respect to 1-dimensional parameters?

Given the following matrices and vectors, I am trying to derive the gradient of equation (1). $t \in R ,\quad S \in R^{N \times N}, \quad y \in R^N, \quad Q = tS $ and $Q$ is invertible $\frac{\...
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2answers
30 views

How to Complete Implicit Differentiation Problem

The problem I am trying to find $\frac{dy}{dx}$ of: $$\sqrt{x+y}= x^4 + y^4$$ I have attempted to solve the problem via the following steps: $x^{1/2} + y^{1/2} = x^4 + y ^4$ $\frac{d}{dx}x^{1/2}+\...
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1answer
16 views

Number of terms for expression using the chain rule

I am given the following: $z = g(u, v)$ $u = u(x, y, t)$ $v = v(x, y, t)$ $x = x(t)$ $y = y(t)$ And I need to find the number of terms for $dz/dt$ (using the chain rule) Drawing out a tree I ...
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1answer
84 views

What's $\frac{\partial}{\partial X} f(X\otimes A)$?

Based on the answer to this question, I wonder how, using the differential notation, one finds $\frac{\partial}{\partial X} f(X\otimes A)$? Assume that $X,A$ are positive definite matrices, and we ...
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1answer
32 views

How to Calculate the gradient of $y=(4-2x^2)^5$ at the point where $x=1$

I think that I would have to use chain rule here and I did and I got $dy/dx=5(-4x) (4-2x^2)^4$ but I have no idea how to use the gradient in this case...like do I expand my brackets or do I just ...
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1answer
22 views

Chain rule (hopefully simple question?)

I am trying to find the following partial derivative. $$\frac {\partial }{\partial Y} U(Y-T(Y))$$ I know I need to use the chain rule, and if the function were simply U(T(Y)) it would obviously be ...
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1answer
26 views

Proving that $\frac{D(s_1 , s_2)}{D(a,b)} \frac{D(a,b)}{D(x,y)} = \frac{D(s_1 ,s_2)}{D(x,y)}$

Using Lagrangian discribtion for incompressible fluid . $$\frac{D(x,y)}{D(a,b)}=\begin{vmatrix} \frac{\partial x}{\partial a} & \frac{\partial y}{\partial a} \\ \frac{\partial x}{\partial b} &...
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2answers
54 views

Finding the velocity of $s$ using the chain rule

Can I find the velocity of $s$ using the chain rule? First I'll explain this diagram (sorry for the crappy quality, I had to draw it on Paint. Also im more of a maths person than an art person). It'...
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1answer
37 views

Derivative of f(x,y) with respect to g(x,y)

In this question Derivative of a function with respect to another function. the chain rule is stated in order to take derivative of single variable function with respect to a function of the same ...
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2answers
28 views

Functions equal implies equal partial derivative.

Suppose there are two surfaces described by the functions $f(x,y)$ and $g(x,y)$. The functions are equal along some line $y(x)$ i.e. $f(x,y(x))=g(x,y(x))$. I realise it's naive to next write $f(x,y(x))...
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2answers
25 views

Question on partial derivative with $w=w(x,y,z) and z=z(x,y)$

I was concerning about $\dfrac{\partial w}{\partial x}$ I am thinking about using the chain rule as z depends on x and get $\dfrac{\partial w}{\partial x}$=$\dfrac{\partial w}{\partial x}$+$\dfrac{\...
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6answers
99 views

Basic proof that $\frac{d}{dx}(\sin(nx))=n\cos(nx)$

Could someone provide a basic proof that $\frac{d}{dx}(\sin(nx))=n\cos(nx)$? I'm using $n$ to be broad, and so this can be searched easier, though if it's easier to provide an example, then replace $n$...
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2answers
47 views

I keep failing to understand what I'm doing wrong - derivatives

I need to find the derivative of this function: $f(x)=x^2(x-2)^4$ I have gone about this two ways: Chain Rule inside a Power Rule $$f(x)=x^2(x-2)^4$$ (I found the derivative of $(x-2)^4$ using the ...
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0answers
36 views

Magnitude of gradient is rotationally invariant

Consider an image with edges, which we can take a gradient of (i.e. by subtracting pixels). Suppose we have a point $(x,0)$ on an edge of the image. Now suppose we rotate this point by an angle ...
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0answers
23 views

How by using rotation matrix to relate polar $\frac{\partial}{\partial \rho}$ , $\frac{\partial}{\partial \phi}$ to Cartesian partial derivatives

How by using rotation matrix to relate the $\frac{\partial}{\partial \rho}$ and $\frac{\partial}{\partial \phi}$ to Cartesian partial derivatives? We do not want to use chain rule. The rotation ...
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1answer
41 views

Confusion about implicit differentiation and chain rule

This silly question comes in two parts. First, I wanted to solve problem 58 in chapter 14.5 of Stewart, 8th edition, which states: Suppose that the equation $F(x,y,z) = 0$ implicitly defines each of ...
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0answers
20 views

Chain rule confusion in defining composite function

I am currently a Calc 1 student, and I have learned the Chain rule for differentiation. The Chain Rule states that [f(g(h(x)))]' = f'(g(h)) * g'(h) * h'. I totally understand this. However, I do ...
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1answer
56 views

Integral of chain rule of total derivative

I am trying to understand how you would 'reverse' the chain rule for a derivative. If I have a function $$f(x(r,\theta),y(r,\theta))$$ then differentiating w.r.t $r$ gives: $$\frac{\partial f}{\...
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6answers
227 views

Proof of the chainrule: is this proof correct and did I use the right notation?

I created this proof of the chainrule. Being a (relative) beginner at math I have a few questions. Is the proof below correct? I was especially in doubt about the use of $h$ on both sides. Is the (...
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1answer
40 views

Chain rule for derivative of a norm

Suppose that $A$ is an $M \times N$ matrix, $x$ is an $N \times 1$ vector and $b$ is an $M \times1$ vector. I want to compute $\frac{d}{dx}||Ax+b||^2_{2}$. According to this link, the answer should ...
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1answer
69 views

Derivative of Euclidean norm

Assume $X$ is a $n$ by $d$ matrix, $\alpha$ is a $n$ by $1$ vector, then $$\frac{d\|X^T\alpha\|^2_2}{d\alpha}=\frac{d\|X^T\alpha\|^2_2}{dX^T\alpha}\frac{dX^T\alpha}{d\alpha}=2\alpha^T X X^T.$$ I was ...
2
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1answer
96 views

Understanding the chain rule in the Wirtinger calculus

The Wirtinger differential operators are defined by: \begin{equation} \frac{\partial}{\partial z} = \frac{1}{2}\left(\frac{\partial}{\partial x} - i\frac{\partial}{\partial y}\right) \\ \frac{\...
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2answers
53 views

Differential on Tangent Spaces and the Chain Rule

From Lee's Intro to Smooth Manifolds I don't see how this is a straightforward application the chain rule. The idea of having different coordinates on the domain and codomain is throwing me off. I ...
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0answers
8 views

How to compute the partial derivative of the Hamiltonian?

The Hamiltonian associate to the Lagrangian $L$ is defined as $$H(p,x) = p\cdot v(p,x)-L(v(p,x),x).$$ We also make the following important hypothesis: suppose that for all $p,x\in \mathbb{R}^n$ the ...
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Question about the derivation of the charecteristics of the Hamilton Jacobi Equation.

I am reading Evan's book on PDE where he mentions the Hamilton Jacobi Equation as follows: $$G(Du,u_t,u,x,t) = u_t + H(Du,x) = 0$$ where $Du=D_xu = (u_{x_1},u_{x_2},...,u_{x_n}).$ Then writing $q=(p,...
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1answer
70 views

Higher total derivatives (Frechét derivatives)

This issue I just cannnot resolve, so I'd highly appreciate your help. Let $a_1, ... , a_n \in \mathbb{R}^k$ with $k$ a natural positive number. If we consider the function $$ W: \mathbb{R}^k \to \...
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5answers
58 views

Antiderivative of $x\sqrt{1+x^2}$

I am attempting this problem given to me, but the answer key does not explain the answer. The question asks me to find the antiderivative of $x\sqrt{1+x^2}$. My attempt: $\frac{d}{dx}f(g(x)) = g'(x)...
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2answers
52 views

Implicit differentiation vs chain rule

Suppose we have the equation $V = \frac{1}{3}\pi r^2h$. Find $\frac{dr}{dh}$. [Chain Rule] We have $\frac{dV}{dh}=\frac{dV}{dr}\cdot\frac{dr}{dh}$. $$ \begin{cases} \frac{dV}{dh}=\frac{1}{3}\pi r^2\...
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0answers
20 views

Algebra partial derivative using chain rule for $f (x, y, z) =\frac{\sin z}{x^2+y^2}$.

Hi i am trying to solve this question $$f (x, y, z) =\dfrac{\sin z}{x^2+y^2}$$ where $$x (r, θ, h) = r \cos θ$$ $$y (r, θ, h) = r \sin θ$$ $$z (r, θ, h) = h$$ first part of the question was to ...
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0answers
26 views

chain rule, $\nabla u\left(\frac{x}{\vert x \vert^2}\right)$

find $\nabla u\left(\frac{x}{\vert x \vert^2}\right)$ where $x \in \mathbb{R}^n$ I am somewhat confused about how to write it out, what I have is (using chain rule) $$D_iu\left(\frac{x}{\vert x \vert^...
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0answers
68 views

A better understanding on application of chain rule.

I came across one problem in my calculus course which I am stuck with for a while. Let $f(x,y)$ posses $n$-th order partial derivatives. Let $x = a + ht $ and $ y = b + ht $ ; $h,k,a,b $ are ...