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Questions tagged [chain-rule]

For questions involving the chain rule in analysis. The chain rule is a special rule to differentiate a composition (chain) of several functions. In single-variable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule.

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Sketchy use of multivariable chain rule under too weak hypoteses

I've found this statement in my real analysis course notes: Let $f: B_r (x_0) \subseteq \mathbb{R}^m \to \mathbb{R}$ ($B_r (x_0) = \{ x \in \mathbb{R}^m : d(x, x_0) < r \}$) be a function such ...
Rick Does Math's user avatar
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32 views

Multivariate Chain Rule with Derivatives in Intermediate Functions

I have a function $$G: \mathbb R^d \times \mathbb R \times \mathbb R^d \to \mathbb R$$ where $d$ is a positive integer and the arguments of $G$ are denoted by $({\bf y}, z, {\bf p})$. I'm denoting ...
Anson's user avatar
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0 answers
27 views

Using the Chain Rule to Switch Between Coordinate Axes in a PDE

I have a system of PDEs that are written in terms of two coordinates: $(x,z)$ which are the usual Cartesian coordinates and $(l,n)$ which are tangential and normal components to some deformable ...
Mjoseph's user avatar
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1 answer
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Why does the transformation of a random variable require the transformation have continuous derivative and the original pdf be continuous?

In Statistical Inference (2e, Casella & Berger), there is the following theorem (p. 51) (emphasis mine): Theorem 2.1.5 Let $X$ have pdf $f_X(x)$ and let $Y = g(X)$, where $g$ is a monotone ...
Kyle L's user avatar
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1 answer
39 views

Chain rule to differentiate vector fields with a connection or Lie derivative

Let $X : M \rightarrow TM$ be a smooth vector field, $f : M \rightarrow \mathbb{R}$ a smooth real valued function and $\gamma : [0,T] \rightarrow M$ a smooth curve in $M$. I am interested in ...
Theo Diamantakis's user avatar
5 votes
2 answers
368 views

Applying the Chain Rule for ($\sin(x) + 1$)

So, according to the chain rule, $$ \frac{d(f(g(x)))}{dx} = f'(g(x)) \cdot g'(x). $$ Now, if we considered $f(x) = x+1$ and $g(x) = \sin(x)$ then: $$ f(g(x)) = \sin(x)+1 $$ In this case, shouldn't ...
Hemavathi Venkatraman's user avatar
0 votes
1 answer
26 views

Multivariable Chain Rule with a scaled Function

Let $f: \mathbb{R}^d \setminus \{0\} \to \mathbb{R}$ be continuously differentiable. For $x \in \mathbb{R}^d\setminus \{0\}$ and $\lambda \in (0,\infty)$, define the function $g_x : (0,\infty) \to \...
supermaxy4's user avatar
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0 answers
11 views

Instance of chain rule in Spivak's Calculus on Manifolds after implicit function theorem [duplicate]

I have a question about a use of the chain rule in Spivak's Calculus on Manifolds. I believe this is just some basic confusion I have with the notation. Let $f:\mathbb R^n\times\mathbb R^m\to\mathbb R^...
psie's user avatar
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2 votes
1 answer
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Double derivative representation in terms of $\theta$?

Ok, this is a thought I had, as we all know that we can write $\mathrm dy/\mathrm dx$ as $\tan(\theta)$ because it's just the slope of the tangent line. Thought I could get the double derivative in ...
cdkw2's user avatar
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117 views

Application of Chain Rule in Proof of Stokes' Theorem

In the proof of Stokes' theorem in "Vector Calculus" by Marsden and Tromba, I noticed that the chain rule is applied selectively. Specifically, the chain rule is not used when expanding the ...
Vova N's user avatar
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Rewriting the second derivative of a function by substitution

I would like to know if the equation $$ \frac{d^2T(x)}{dx^2} = \frac{1}{2}\cdot\frac{d}{dT}\left(\frac{d}{dx}T(x)\right)^2\quad(1) $$ is true for a general function T(x). The function T(x) describes ...
Emann's user avatar
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58 views

Chain rule with functional derivatives?

I'd like to make the functional derivative of the functional $S[\phi(x)]$ with respect to the Fourier transform $\widetilde{\phi}(p)$ such that $$\phi(x)=\int\frac{d^{d}p}{(2\pi)^{d/2}}e^{ip\cdot x}\...
Filippo's user avatar
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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
3 votes
1 answer
81 views

Wirtinger Matrix Derivative Chain rule

I'm trying to compute the matrix Wirtinger derivative $$\frac{\partial (f\circ g)(Z)}{\partial Z}$$ where $g(Z) := B(A Z-Z A)$ and $f(g(Z)):= \mathrm{Tr}\left(\sqrt{g(Z)^* g(Z)}\right)$. Here $Z$ is a ...
Aritra Das's user avatar
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1 vote
1 answer
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Why is the numerator-layout Jacobian transposed in backpropagation calculation?

In the derivation of the backpropagation algorithm in Neural Network Design by Hagan et al., we consider the derivative of the scalar-valued sample loss function $\hat{F}$ with respect to a vector of &...
aas's user avatar
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2 answers
71 views

Counter example: if each function of a composite function being not differentiable at a point, then the composite function is also not differentiable

I was reading counter example in calculus book. And stuck one problem: If a function $g(x)$ is not differentiable at $x=a$ a and a function $f(x)$ is not differentiable at $g(a)$, then the function $...
falamiw's user avatar
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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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2 votes
1 answer
59 views

Chain rule for Clarke-derivatives

The Clarke-gradient is often introduced to extend ideas from convex analysis to non-convex functions, see [Clarke, Sec 2.1]. In particular, given $f:\mathbb{R}^n\rightarrow \mathbb{R}$ Lipschitz in $x$...
Bazinga's user avatar
  • 183
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1 answer
57 views

Can $F(x)=g(f(x))$ be differentiable at $x=\alpha$ if f and g are not at $x=\alpha$

is $F(x)=g(f(x))$ always non-differentiable at $x=\alpha$ if: a) f is differentiable at $\alpha$ and g is not différentiable at $f(\alpha)$ b) f is not différentiable at $\alpha$ and g is ...
edster101's user avatar
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65 views

Applying chain rule/product rule to $\frac{d}{d\tau} \left( \frac{d}{dt} \theta(t(\tau)) \, \frac{d}{d\tau} t(\tau) \right)$

How can I apply the chain rule to following function? $$ \frac{d}{d\tau} \left( \frac{d}{dt} \theta(t(\tau)) \, \frac{d}{d\tau} t(\tau) \right) $$ The right way to rewrite the above expression is by ...
Federica Guidotti's user avatar
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35 views

Generalized chain rule for multivariable functions

Let x(t) = $(x_1(t),\dots,x_m(t))$ and let $f: \mathbb{R}^m \to \mathbb{R}$ be a multivariable function. Then let $\hat f$(t) = f (x(t)). I know that $$\frac{d\hat f(t)}{dt} = \sum_{h=1}^m \frac{\...
Davide Masi's user avatar
2 votes
3 answers
186 views

How do I calculate the derivative of a composition $R^{n} \rightarrow R^{n \times n} \rightarrow R^{n}$?

I am having problems calculating the derivative of a function. Let $C:\mathbb{R}^{n \times n} \longrightarrow \mathbb{R}^{n}$ with $C(M) = (I - M)^{-1}(I + M)x_0$ for (I - M) invertible ($x_0 \in \...
Donnie's user avatar
  • 23
1 vote
0 answers
232 views

About the chain rule of the exponential entropy

In the paper unifying framework of information measures the conditional exponential entropy (see equation 29) is defined as: $\mathcal{E}_{\alpha}(X|Y) = E_y\left(\int_{\mathbb{R}} f^{\alpha}(x|y)\,...
Upax's user avatar
  • 2,115
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1 answer
30 views

Finding a derivative for option pricing

I am trying to find the derivative of the expression below wrt $S$: $$ e^{-dt}N(x) $$ where $x$ is defined as: $$ \frac{\ln({\frac{S}{K}})}{\sigma\sqrt{t}} $$ and $N(x)$ is the cumulative of the ...
Nikowhy's user avatar
  • 31
0 votes
1 answer
30 views

Chain rule with a variable held constant (or along a constant surface) [duplicate]

Say we have a Cartesian function, $f(x,y)$ and we move to a polar coordinate scheme $g(r,\theta)$, $$g(r,\theta) = f(x,y) \\ \therefore x = x(r,\theta), y = y(r,\theta)$$ Just to lay out the ...
Researcher R's user avatar
2 votes
1 answer
79 views

Strange usage of chain rule. Can anyone explain why this derivation was done this way?

There are 2 issues I have with the way this was done. The first was how chain rule was used in the (1.35), and the second was how chain rule was used in (1.36). It all seems so counterintuitive. For (...
Researcher R's user avatar
1 vote
2 answers
133 views

Finding derivative with two functions sharing the same independent variable

Find $\frac{dy}{dx}$ , provided that $x=f(t)$, $y=g(t)$, $f$ and $g$ are both differentiable, and $y$ is also a differentiable function of $x$, and $f$ is bijective. already have: $(h\circ i)'=(h'\...
joggingrat's user avatar
0 votes
0 answers
56 views

Chain rule for integrator function with Riemann-Stieltjes integral

I have a random variable $X$ with distribution function $F$. I am interested in evaluating the integral $$ \int g(x) \beta'(F(x)) F(dx) $$ where $\beta$ is smooth and monotone and $g$ is such that the ...
Masanja M.'s user avatar
0 votes
1 answer
52 views

Change of coordinates on $\nabla(h\circ\varphi^{-1})$ where $h,\varphi:\mathbb{R}^n\to\mathbb{R}^n$

Say I have a smooth vector-valued function $h:\mathbb{R}^n\to\mathbb{R}^n$ and a smooth diffeomorphism $\varphi:\mathbb{R}^n\to\mathbb{R}^n$. Consider the gradient of the composition $h\circ\varphi^{-...
Stuck's user avatar
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33 views

The chain rule for partial derivatives

My goal is to understand a statement about partial derivatives made in the appendix of a textbook about manifolds. From the book: Let $U \subseteq \mathbb{R}^n$ and $\tilde{U}\subseteq \mathbb{R}^m$ ...
Maple's user avatar
  • 13
1 vote
1 answer
71 views

Prove an equality of derivatives [duplicate]

Suppose $F:\mathbb{R}^3\to\mathbb{R}$ is of class $C^1$ and there exists $n\in\mathbb{R}$ s.t $F(tx,ty,tz)=t^nF(x,y,z)$ for all $t>0$ and for all $(x,y,z)\in\mathbb{R}^3$. Prove that $$x\dfrac{\...
lee max's user avatar
  • 249
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0 answers
38 views

To prove $f_2\circ f_1$ is differentiable on $U$, can we weaken this assumption?

I am reading "Analysis on Manifolds" by James R. Munkres. Theorem 7.1. Let $A\subset\mathbb{R}^m$; let $B\subset\mathbb{R}^n$. Let $$f:A\to\mathbb{R}^n\,\,\,\,\,\,\text{and}\,\,\,\,\,\,g:B\...
佐武五郎's user avatar
  • 1,106
2 votes
2 answers
89 views

Chain rule and differentiability of $|x|^2$

Going through Thomas Calculus, question 90 in the chapter on Chain Rule: Suppose that $f(x) =x^{2}$ and $g(x) =|x|$. Then the composites $$ ( f\circ g)( x) =|x|^{2} =x^{2} \ \ \ \ \ \ and\ \ \ \ \ ( ...
Stanislav Bashkyrtsev's user avatar
1 vote
1 answer
47 views

Compute the derivative using chain rule

Let $z=f(u,v)$ where $u=xy$ and $v=\frac{x}{y}$ such that $f$ is second differentiable. Compute $\tfrac{{\partial}^2z}{\partial x\partial y}$ and $\tfrac{{\partial}^2z}{\partial x^2}$. My attempt for ...
user avatar
1 vote
0 answers
27 views

Given the wave equation, show $u_{yw}=0$

Consider the wave equation: $\alpha^2 u_{xx}=u_{tt}$. I am told to let $y=x-\alpha t$ and $w=x+\alpha t$ and use chain rule but I'm not confident in my attempt. I got $x=\frac{y+w}{2}$ and $t=\frac{w-...
PeakyBlaze7788's user avatar
3 votes
1 answer
102 views

Example implementing the Chain Rule in a textbook by Charles Chapman Pugh

I am asking for help interpreting an example in a textbook. The author gives two functions from different dimensions of Euclidean space, and he precisely describes the image of arbitrary elements ...
user74973's user avatar
  • 706
2 votes
1 answer
62 views

differential chain rule

I'm trying to follow some calculus lecture notes and I can reproduce an argument for the sum and product rule. Can I make an argument like this for the chain rule in a similar style, using the ...
andi's user avatar
  • 109
0 votes
1 answer
42 views

Derivative of $ |u|^\gamma $ with $ \gamma>1 $

I have a doubt in the proof of Gagliardo-Nirenberg-Sobolev inequality that is found in Evans' book "Partial Differential Equations" (Theorem 1, pages 277-279). In this part of the proof, $ \...
jom's user avatar
  • 33
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0 answers
126 views

Backpropagation: Chain Rule for Matrix Exponential?

Recent linear state-space model papers like Mamba often use matrix exponential to discretize the system. They initialize the system in a continuous-time regime, and discretize it to run it like a ...
lostintimespace's user avatar
0 votes
2 answers
72 views

Proving that Jacobian of Composition is equal to Composition of Jacobians using epsilon-delta

Let us have functions $\mathrm{f}: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $\mathrm{g}: \mathbb{R}^m \rightarrow \mathbb{R}^k$ such that $\mathrm{f}$ is differentiable at some point $\mathrm{a} \in ...
Timothy Leong's user avatar
1 vote
0 answers
52 views

Doesn't Jacobi formula immediately follow from the chain rule

It's easy to see that the derivative of $detA$ is $adj(A)^T$ applied to $A(t)$ we have $adj(A(t))^T$ and the derivative of $A(t)$ is simply $A'(t)$. Now "multiplying" all this together with ...
kingW3's user avatar
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1 vote
1 answer
61 views

Can you square root both sides of an implicit equation to get the derivative? [closed]

Suppose you have $x^2=(4x^2y^3 + 1)^2$, can you square root both sides of the equation to make it simpler?
mathingishard's user avatar
0 votes
1 answer
58 views

What kind of product operator appears in the chain rule involving a derivative wrt a matrix?

Let $$y = Ax$$ $$z = y^Ty$$. Per the chain rule, $$\frac{\partial{z}}{\partial{A}} = \frac{\partial{y}}{\partial{A}} \frac{\partial{z}}{\partial{y}} $$ Since, $$\frac{\partial{z}}{\partial{A}} = \frac{...
Tomek Dobrzynski's user avatar
3 votes
2 answers
113 views

"Correct" way of seeing that $\frac{\partial}{\partial A} f(AB) = \frac{\partial f(X)}{\partial X} B^T$, where $X = AB$

Lemma: Let $A \in \mathbb{R}^{m\times n}$ and $B \in \mathbb{R}^{n\times k}$ be matrices, and let $f:\mathbb{R}^{m\times k} \to\mathbb{R}$ be a differentiable function. Let $X = AB$. Then $$ \frac{\...
Sebastian Monnet's user avatar
0 votes
0 answers
61 views

Simple matrix calculus, and yet I am struggling to understand

Here is my problem: We have $\mathbf{D} \in \Re^{m n}$, $\mathbf{W} \in \Re^{m q}$, and $\mathbf{X} \in \Re^{q n}$. Furthermore, $\mathbf{D} = \mathbf{W}\mathbf{X}$. (NOT an element wise ...
wrek's user avatar
  • 485
1 vote
1 answer
92 views

Using chain rule to find $\frac{dy}{dx}$ of an inverse sine, got negative of the actual solution

Problem Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$, where $y = \sin^{-1}\biggl(\dfrac{2x}{1+x^2}\biggr)$ Given solution $\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{2}{1+x^2}$ My approach I got the ...
Cinverse's user avatar
  • 181
1 vote
1 answer
76 views

Two ways of solving the second derivative

Here is the situation: say $f(u)=y=u^2$, and $g(x)=u=x^2$ and we want to find $\frac{d^2y}{dx^2}$. I have two ways of solving this: Substituting $u=x^2$ and writing $y$ as, $$y=(x^2)^2=x^4$$ $$\frac{...
Jeffy James's user avatar
1 vote
2 answers
119 views

Stuck on proving the Hessian chain rule when $h(x) = g(f(x))$ is a real valued function.

I wish to prove the chain rule property in the above screenshot. However, I am not able to produce the outer-product. Let $\nabla$ be the gradient operator. Then I know for $h(x) = g(f(x))$ as shown ...
Olórin's user avatar
  • 5,473
1 vote
1 answer
127 views

Derivative of real-valued function that takes a matrix

I want to compute the partial derivative of a real-valued function that takes matrices as argmuents. The function has the form $$F(x,y,z) = ||g(x) \odot (S \cdot y) - z||,$$ where $x, y, z, S \in \...
lfische's user avatar
  • 11
0 votes
3 answers
41 views

How to solve this second derivative?

I want to solve the derivative $\frac{d^{2}\psi}{dx^{2}}$ for $\psi(x) = Ae^{iu(x)}$. Here, A is a constant, $u = u(x)$ and $i$ is the complex number. The answer for this has been given as: $i \frac{d^...
user374355's user avatar

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