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Questions tagged [chain-rule]

For questions involving the chain rule in analysis.

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Rule for derivation: $d^n/dt^n$, where $t = -ln(x)$.

I'm trying to show that $$ (-1)^{n+1} \frac{d^n}{dt^n}(1 - e^{-t})^\alpha = -(x \frac{d}{dx})^n(1-x)^\alpha$$ where $x = e^{-t}$, $\alpha > 1, n > 0$. If I understand correctly, $$ \...
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Finding a relation between functions according to known constraints

I am solving a problem on geodesics with ideas from General Relativity and got stuck with one step. The simplified version is the following: With notations $$\dot{x}\equiv \frac{dx}{dt}, \quad \...
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0answers
27 views

how to calculate $z'(y) $ near $ y = 0$. of the curve $g(t) = (e^t + t,~t^2+3\sin(t)~,t^4+t+1)$

$ Let ~g(t) = (e^t + t,~t^2+3\sin(t)~,t^4+t+1)$ such that $g(0) = (1,0,1)$ calculate$~~z'(y) $ near $ y = 0$. how do i solve this using implicit function theorem ? it was an exam question ,...
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37 views

What is the derivative of $\operatorname{trace}(XCP(XC)^T)$?

I am really stuck at calculating $\frac{d\operatorname{trace}(XCP(XC)^T)}{dC}$ where $P \in R^{r\times r}$, $X \in R^{m\times n}$ and $C \in R^{n\times r}$ . Do I need to recall $A=XC$ and then apply ...
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1answer
33 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 ...
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2answers
44 views

Technical question about the proof of the chain rule

In Spivak calculus proof of the chain rule, which is essentially the same as this proof, Spivak mentioned that the continuity of $\phi(h)$ is the crux of the whole proof I'm not sure I see why ...
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32 views

Can the product rule be used in thie manner?

I'm trying to understand the proof to Euler's formula and I keep getting stuck on one of the steps. We start with: $${\displaystyle e^{ix}=r(\cos \theta +i\sin \theta ).}$$ We will then assume that ...
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39 views

Question on Partial derivatives in inverse function changing from coordinate systems

I would like to ask a question about partial derivatives in the context of Rotations of coordinate systems. Say we have a coordinate system (unprimed) and its rotated version (primed). If the ...
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1answer
30 views

Contradict the chain rule [closed]

Let $$f ( x , y ) = \left\{ \begin{array} { c l } { \frac { x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } \text { if } ( x , y ) \neq ( 0,0 ) } \\ { 0 \quad \text { if } ( x , y ) = ( 0,0 ) } \end{array} \...
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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 ...
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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 ...
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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 ...
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A question related to Chain rule

Suppose that the dependent variables $z$ and $w$ are functions of independent variables $x$ and $y$, defined by the equations $f(x,y,z(x),w(y))=0$ and $g(x,y,z(x),w(y))=0$, where $$f_zg_w-f_wg_z=1.\...
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1answer
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joint probabilities chain rule: does the p(a, b, c) equals p(c,b,a)

Hi I am studying joint chain prob chain rule. I found that most of mathematical form is as following: but some is like as following: I can understand clear the second form which using the joint and ...
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2answers
32 views

Time derivative of scalar function that takes vector as argument

Let's say I have scalar function $\phi$ that is function of some vectors $\vec{\bf{p}}$ and $\vec{\bf{r}}$ such that $\phi = \phi(\vec{\bf{p}}-\vec{\bf{r}})$, also vector $\bf{r}$ is function of time, ...
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1answer
49 views

Chain rule, two different interpretations, two different results

I have a function $f:\mathbb{R}^2 \to \mathbb{R}$. So we can write $f(x,y)$ where $x$ and $y$ are independant variables. Now we define the function $g(u,v) = f(u+v, uv)$. Using the chain rule we ...
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2answers
50 views

Abuse of notation in the chain rule

I have a function: $f: \mathbb{R}^p \to \mathbb{R}^n$. Now let's define the functions $x_i : \mathbb{R}^p \to \mathbb{R}$, and hence we can define the function $\phi : (u_1,..., u_p) \to (x_1(u_1,......
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1answer
25 views

Find $\frac{\partial}{\partial p}$ and $\frac{\partial}{\partial q}$ of $U(f(p) + q^2, g(q)^2)$

Here's my question: Suppose $U(x_1, x_2)$ is a differentiable function. Suppose $f(p)$ and $g(q)$ are differentiable functions that depend only on $p$ and $q$ respectively. Find $\frac{\partial}{\...
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1answer
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Compute the gradient of $f(x)=\|\text{diag}(x)\|$ with the chain rule

Consider the function $f:\mathbb{R}^n\to\mathbb{R}$ given by $f(x)=\|\text{diag}(x)\|$, where $\text{diag}(x)\in\mathbb{R}^{n\times{n}}$ is the diagonal matrix with diagonal entries $x_1,x_2,\dots,x_n$...
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2answers
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Chain rule for derivatives and simplification question.

I am trying to find the derivative of the function $$-\frac 13 (e^x −1)^2 +(e ^x−1)+ \frac 15$$ I can only get as far as: $$e^x - \frac{ 2(e^x - 1)e^x}{3}.$$ The answer I am looking for is $$-\...
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1answer
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Why can't I reduce the total differential?

I have encountered the following equation: $g: \mathbb{R}^m \rightarrow \mathbb{R}$ $u: \mathbb{R}^n \rightarrow \mathbb{R}^m$ $z = g(\mathbf{y})$, $\mathbf{y} = u(\mathbf{x})$ then using ...
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2answers
26 views

Prove that product of two matrices is $I$

The pair of variables $(x, y)$ are each functions of the pair of variables $(u, v)$ and vice versa. Consider the matrices: $$ A=\left(\begin{matrix} \frac{\partial{x}}{\partial{u}}& \frac{\...
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3answers
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Showing solution satisfies PDE (chain rule help)

We have $$u(x, t) = g(x-at, 0) + \int_0^t f(x+a(s-t), s)ds$$ and want to show it satisfies the PDE $$\partial_t u + a \cdot Du = f$$ The solution given goes as follows $$\partial_t u + a\cdot Du = - ...
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35 views

Calculus chain rule clarification

Chain rule clarification: Please have a look at the picture before continuing to read my query as this picture provides the context. On the fourth line of working (the one before the final simplified ...
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1answer
51 views

Applying the chain rule on vectors and matrices

I need to find $\frac{dy}{dx}$ for the following y = $||A^Tx - b||_2^2$ where $A \in R^{3x3}, b \in R^{3x1}, x \in R^{3x1}, y \in R,$ and $||.||_2$ is the euclidean norm so for example $||z||_2^2 = ...
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1answer
50 views

Matrix derivatives, problem with dimensions

I'm trying to find a derivative of function: $$L = f \cdot y; f = X \cdot W + b$$ Matrices shapes: $X.shape=(1, m), W.shape=(m,10), b.shape=(1, 10), y.shape=(10, 1)$ I'm looking for $\frac{\partial ...
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Multivariate Chain Rule with a Function of one of the Variables

I'm working through a particular problem that is giving me inconsistent results given the way I approach the problem, and I would like to confirm that I am appropriately understanding the chain rule ...
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4answers
67 views

Showing $\ddot{x} = \frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{2} \dot{x}^2)$

In Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers one of the very first stated equations are, as in the title of the question, $$ \ddot{x} = \frac{\mathrm{d}...
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38 views

Partial differentiation chain rule, differential operator?

We are given the function \begin{equation} V(x,y)= f(s)+g(t) \end{equation} with s=x+y and t=x+0.5y. How can I calculate $V_{xx}$ and $V_{yy}$? I have calculated $V_{x}$ and $V_{y}$ but I do not know ...
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1answer
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Chain Rule in Multiple Dimensions

Let $x$ be a map, $x:\mathbb{R}\rightarrow\mathbb{R}^n$. Let $V$ be a map, $V:\mathbb{R}^n\rightarrow\mathbb{R}$. Then the derivative $$\dfrac{d}{dt}V(x(t))=\nabla V(x(t))\cdot x'(t)=\langle\ \nabla ...
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2answers
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Confusion on multivariable partial differentiation from textbook

I am very confused about a use of the partial differential multivariable chain rule. The textbook first describes the formula: $\frac{\partial f}{\partial s} = \frac{\partial f}{\partial x}\frac{\...
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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
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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
41 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
64 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
39 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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24 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
30 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
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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
51 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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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
22 views

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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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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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
76 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
49 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
97 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
29 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 ...