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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35 views

Chain rule notation

For the part $\frac{\partial u}{\partial x}$, do we do $\frac{\partial u(x,y)}{\partial x}$ first, then evaluate at the point $(x,y)=(x,y(x))?$ For the part $u_{x}$, do we do $\frac{\partial u(x,y)}{\...
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25 views

$t$ derivative of Kirchhoff's solution

I know that the solution to the PDE \begin{align*} u_{tt} - \Delta u = 0, \quad \mathbb{R}^3\times[0, \infty)\\ u(x, 0) = 0, \quad x \in \mathbb{R}^3\\ u_t(x, 0) = g(x), \quad x \in \mathbb{R}^3 \end{...
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2answers
152 views
+150

Geometric proof of chain rule with the derivative of $\sin(2x)$

I'm following this post https://math.stackexchange.com/a/2169/612996 as my example and I've figured out how it works for $\sin(\theta)$, During my first try: I keep on missing the factor of $2$ when ...
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3answers
51 views

Total derivative of vector function

Let's asume that I have a (vector) function $f \space (s,t,u,v):\mathbb{R}^4 \rightarrow \mathbb{R}^n.$ I would like to calculate: $\frac{d}{dt}\biggr|_{t=t_0} f(t,t,t,t).$ Intuitively: This should ...
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2answers
42 views

Understanding the chain rule for differentiation operators

Suppose I want to transform a partial derivative operator from spherical to Cartesian coordinates. I have found the following relation based on the chain rule here: $$ \frac{\partial }{\partial \theta ...
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1answer
61 views

Chain rule proof confusion

Here is a common informal proof for the chain rule: If $S(a)=f(g(x))|_{x=a}$, then $S'(a)$ is given by \begin{align} \lim_{x \to a}\frac{S(x)-S(a)}{x-a}&=\lim_{x\to a}\frac{f(g(x))-f(g(a))}{x-a} \\...
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1answer
39 views

First Variation of $L_2$ with Linear operator

Let $\Omega \subset \mathbb{R}^2$ be open and bounded, $P:C^1(\Omega) \to L_2(\mathbb{R})$ be a linear and bounded operator. I want to calculate the first variation of $\|Pu - f\|^2_{L_2}$, $u \in C^1(...
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1answer
30 views

Decomposition of a function and chain rule.

This question is about the basic chain rule (and I think of it when I read about calculation of variation in defining distance in manifold using usual Riemannian metrics) and is related to the another ...
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78 views

Is this a good justification for why integration by substitution works? [duplicate]

It is a well known fact that $$ \int f'(g(x))g'(x)dx=f(g(x))+C $$ This follows directly from the chain rule. However, sometimes it is easier to perform the substitution $u=g(x)$: \begin{align} u&=...
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65 views

Are we allowed to apply the chain rule here?

Let $d\in\mathbb N$ and $U\subseteq\mathbb R^d$ be open $\tau>0$ and $T_t$ be a $C^1$-diffeomorphism from $U$ onto an open subset of $\mathbb R^d$ for $t\in[0,\tau)$ with $T_0=\operatorname{id}_U$ ...
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1answer
35 views

Show total differentiability of $\phi: \mathbb R^n \to \mathbb R^n, x \mapsto \varphi(\lVert x\rVert_2) x$ where $\varphi$ is differentiable

Let $\varphi$ be differentiable. Show that $$\phi: \mathbb R^n \to \mathbb R^n, x \mapsto \varphi(\lVert x\rVert_2) x$$ is (total) differentiable where $x \neq 0$. How can I show this? I know that $\...
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1answer
61 views

stuck with how this ordinary differential equation changes in new coordinates?

I am studying ordinary differential equations. There are some solved examples in my book to learn the material from. For this one below however, I do not know how this part of the solution is derived. ...
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1answer
42 views

Understanding the multidimensional chain rule

I have some trouble in understanding the multidimensional chain rule. For differentiable functions $f,g$, defined by $f: U \to V$, $g:V\to\mathbb K^n$ where $U \subseteq \mathbb R^d$, $V \subseteq \...
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An exercise regarding the computation of partial derivatives and applying the Chain Rule

I came across the following partial derivative exercise: Exercise: Let $F(x,y) = f(x^2+g(x+2y))$, where $f$ and $g$ are differentiable functions of one variable. Given that the equation $F(x,y)=0$ ...
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63 views

Prove that $y'$ is a function of $y$

First, some context: This occurred to me while I was learning how to do order reduction for $2^{nd}$ order ODEs of the form $F(y, y', y'') = 0$. Apparently, I'm supposed to set $p = y'$, say that $y'' ...
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1answer
80 views

Why can't one cancel partial differentials?

I have a question about the below formula: $$\frac{dz}{ds} = \frac{dz}{dx} \cdot \frac{dx}{ds} + \frac{dz}{dy} \cdot \frac{dy}{ds}$$ Ok. I understand what this means. Small change of s makes small ...
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1answer
41 views

how to use the chain rule in a multivariable function?

Problem: Let the function $f(x,y)=(x^2+y^2)\sin(x)$ where $x=r^2e^s$ and $y=rs$ Using the chain rule compute $\frac{\partial f}{\partial r}$ and $\frac{\partial f}{\partial s}$ and then compute $\...
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1answer
13 views

Subgradient of argmax and chain rule

Let $\mathcal{X} \subset \mathbb{R}^n$ and $c \in \mathbb{R}^n$. Moreover, define $$ f(c) := \max_{x\in\mathcal{X}} \ x^\top c \quad \text{and} \quad \bar{x}_c := \text{arg}\max_{x\in\mathcal{X}} \ x^\...
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1answer
54 views

Confusion about matrix derivative/chain rule

I have a vector Z of which depends on time. I am looking to find $\ddot\sigma(Z)$, the second time derivative of $\sigma$. $$Z = \begin{bmatrix} z_1\\ z_2\\ z_3\\ \end{bmatrix}$$ And $\sigma$ is ...
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How does computation of derivative with multiple parameters work in detail?

I am studying computer science and read the article https://medium.com/datadriveninvestor/how-do-lstm-networks-solve-the-problem-of-vanishing-gradients-a6784971a577 (sub section "The error ...
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1answer
48 views

Why can't the “chain rule” of derivatives be used to differentiate 3sin(x)?

My understanding is that functions of the form $f(g(x))$ can be differentiated using the "chain rule", where $$\frac{d}{dx}f(g(x)) = f'g(x) \cdot g'(x)$$ I was trying to apply that logic to ...
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1answer
34 views

Implicit Function Theorem on a composition of functions [duplicate]

Let $f(x,y)$ be a $C^1$ function where $f(0,0)=0$. What conditions on f guarantee $f(f(x,y),y))=0$ can be solved for $y$ as a $C^1$ function of $x$ near $(0,0)$ Hello, I've been stuck on this problem ...
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57 views

Why cannot cancel terms $\frac{\partial \rho}{\partial t}$ for both side of the equation, chain rule

Suppose I have a function $\rho(x(X,t),t)$, and I perform the partial derivative based on chain rule, what I have is $\frac{\partial \rho}{\partial t} = \frac{\partial\rho}{\partial x}\frac{\partial x}...
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41 views

Extending [a,b] to the real line by a conform mapping in ODEs

- I have an ODE on [a,b] as follows. - I want to extend to [a,b] to the real line by a conform mapping $\psi(t)$. - Substituting the equations to an original ODE, we have a new ODE on $(-\infty,\...
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1answer
81 views

Gradient of complex-valued function with respect to real and imaginary components

Let $J(\mathbf{z})$ be a complex-valued (scalar) function where $\mathbf{z}\in \mathbb{C}^n$, and write $\mathbf{z} = \mathbf{x} + i \mathbf{y}$ for real vectors $\mathbf{x}, \mathbf{y} \in \mathbb{R}^...
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1answer
47 views

How does the square root disappear when differentiating $y=\frac{\sqrt{2x^2}}{\cos x}$?

Finding the derivative of $$y=\frac{\sqrt{2x^2}}{\cos x}$$ I am going through the steps and having trouble using the quotient rule. I have seen the final answer, and I've had no trouble using the ...
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Chain rule for subdifferentials of nonconvex functions

I have two functions: one of them $h\colon\mathbb{R}^n\to\mathcal{S}$ is smooth, but not necessarily convex, and the other $g\colon\mathcal{S}\to\mathcal{S}$ is convex, non-expansive, and not ...
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3answers
83 views

Clarifying the chain rule terminology in differential geometry calculuations

Let $M$ be a manifold and $f:M\to\mathbb{R}$ a smooth function on it. Let $p\in M$ have the coordinates $\{x^i\}$ under the chart $(U,\phi)$. Finally, let $\gamma:I\to M$ be a curve ($I$ is an open ...
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13 views

If $g$ is smooth and $f$ is convex non-smooth, then $\partial f(g(x))\cdot Dg(x)\subseteq \partial (f\circ g)(x))$?

Let $g\colon\mathbb{R}^n\to\mathbb{E}$ be a smooth function and $f\colon\mathbb{E}\to\mathbb{E}$ be a convex function (for my purposes, a projection onto a convex set), where $\mathbb{E}$ is an $m$-...
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18 views

Chain rule explanation

I have a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and a function $\phi: \mathbb{R}^{n-1} \rightarrow \mathbb{R}$ $f$ is defined as $f(x_1,...,x_n) = f(x_{1},...x_{i-1},\phi_{i}(x_{1},...,x_{...
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1answer
49 views

How to show $\Delta f(r) = f''(r) + \frac{2}{r} f'(r)$?

I'm trying to solve the following problem: Given $\vec{r} = (x,y,z)$ , $r= \lVert \vec{r} \rVert$ and $f:\mathbb{R} \to \mathbb{R}$ a twice differentiable function, show that $$ \Delta f(r) = f''(r)...
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5answers
93 views

Am I misapplying the chain rule when differentiating $x^{5x+7}$ with respect to $x$?

The problem I am attempting to solve is: \begin{align} y=x^{5x+7} \\ \text{Find $\frac{dy}{dx}$} \end{align} Here is my working so far: $$\begin{align} \text{let }u &= 5x+7 \\ \frac{dy}{dx}&=\...
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1answer
42 views

Does there exist a gradient chain rule for this case?

My question comes from this article in Wikipedia. I noticed that there is a chain rule defined for the composition of $f:\mathbb{R}\to\mathbb{R}$ and $ g: \mathbb{R}^n \to \mathbb{R}$ given by $$ \...
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1answer
30 views

Finding extreme values using chain rule in multivariate function

We are given the function $$ F(x,y) = (x^2 + y^2)^2 - 2(x^2 - y^2) $$ with the condition $F(x, y(x)) = 0$ for $$ y: (0, \sqrt{2}) \to \mathbb{R}, x \mapsto y(x) $$ The objective is to compute all ...
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3answers
44 views

Why is this reasoning wrong?

Let $w = f(x,y,z), z = g(x,y)$, then by chain rule $$\frac{\partial w}{\partial x} = \frac{\partial x}{\partial x}\frac{\partial x}{\partial x} + \frac{\partial w}{\partial y}\frac{\partial y}{\...
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1answer
20 views

Need help understanding step in the proof of multivariable chain rule

Theorem: Let $k\in\mathbb{N}$, $x_{0},x\in A\subseteq\mathbb{R}^{n}$, $x_{0}\neq x$ and $f:A\to\mathbb{R}$ satisfy that $\partial^{\alpha}f$ exists and is differentiable on $L=\{(1-t)x_{0}+tx\in\...
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21 views

The temperature of a metal plate is given by $f(x,y)=\frac{150}{\sqrt{x^2+y^2+1}}$. Find the R.O.C of the temperature at the point $(8,4)$…

The temperature of a metal plate is given by $f(x,y)=\frac{150}{\sqrt{x^2+y^2+1}}$. Find the rate of change of the temperature at the point $(8,4)$, in the direction towards the point $(7,2)$ Right ...
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1answer
17 views

Consider $w=2x+y^2z$ with $x=p\cos\theta\sin\phi$, $y=p\sin\theta\cos\phi$, $z=p\cos\theta$.

Consider $w=2x+y^2z$ with $x=p\cos\theta\sin\phi$, $y=p\sin\theta\cos\phi$, $z=p\cos\theta$. Find the partial derivatives $w_p$,$w_\theta$, $w_\phi$, each in terms of only the independent variables $p,...
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13 views

Let $z=f(x,y), x=x(u,v),y=y(u,v)$ with $x(3,0)=5, y=(3,0)=2$. Calculate $z_v(3,0)$ using the information below.

Let $z=f(x,y), x=x(u,v),y=y(u,v)$ with $x(3,0)=5, y=(3,0)=2$. Calculate $z_v(3,0)$ using the information below. $f_x(3,0)=r$, $f_y(3,0)=b$, $x_u(3,0)=4$,$y_u(3,0)=q$ $f_x(5,2)=a$, $f_y(5,2)=p$, $x_v(3,...
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2answers
19 views

Let $z=(x+y)e^y$ with $x=7t$ and $y=1-t^2$. Find $\frac{dz}{dt}$ as a function of $t$.

So this is a chain rule question. So I drew out that I need $\frac{dz}{dt}=\frac{dz}{dx}\frac{dx}{dt}+\frac{dz}{dy}\frac{dy}{dt}$. I'm not sure if this is correct, but from here I computed that $\frac{...
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1answer
62 views

Multivariable Chain Rule Notation Clarification?

I am unsure as to what notation is allowed and what is not, even though I can describe the the partial derivatives in words, I sometimes apparently get the notation incorrect. To fully compare and ...
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6 views

Using the chain rule to chain between differential parameters (i.e. independent variables)

I want to ask a question about the chain rule. I know that for the chain rule I can differentiate a product of two different terms together: $$\frac{d}{dx}\left(ln x \right) = \frac{1}{x}$$ This ...
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2answers
35 views

Chain rule for matrix-vector composition

Suppose $A(t,x)$ is a $n\times n$ matrix that depends on a parameter $t$ and a variable $x$, and let $f(t,x)$ be such that $f(t,\cdot)\colon \mathbb{R}^n \to \mathbb{R}^n$. Is there a chain rule for $...
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6answers
104 views

Differentiability of $\cos \lvert x\rvert$

I know that $f(x) = \cos\lvert x\rvert$ is differentiable at $x=0$ and I know what its graph looks like. But if I differentiate $f(x)$ with respect to $x$ , I will have to apply the chain rule i.e, $\...
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0answers
38 views

Chain rule applied in one point

Let $g(u, v)=e^u + \sin v$ and $f(x, y, z)=(xy, xz)$ . Calculate $D(g o f)$ in $(0,1,0)$ applying chain rule. So here is what I have done: $$\frac{\partial g}{\partial x} = \frac{\partial g1}{\partial ...
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0answers
61 views

Mistake in proof of chain rule?

I saw proof of chain rule and I must say that I don't understand one key step. Here is the proof: Consider function $f(x)$ and tangent $t.$ Let $\frac{dy}{dx}$ be slope of a tangent line in the ...
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1answer
22 views

Help with differentiation chain rule with tensors in backpropagation

Say, we're given $N$ feature vectors $\mathbf{x}_i \in \mathbb{R}^{D \times 1}$ and assembled into a matrix $X \in \mathbb{R}^{D \times N}$. We also have a matrix $W \in \mathbb{R}^{D \times D}$, $W = ...
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1answer
28 views

Behaviour of the composition $g \circ f(x)$ around $x=5$ as $f(x) = x^2 - 3x - 10$ and $g(x) = |x|$

I'm a university student taking a real-analysis course and I have been given the following questions regarding the behaviour of the above composite function around $x=5$ Firstly it asks to explain ...
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1answer
15 views

$F(\kappa, \nu)$ with $\kappa = \frac{x_1}{x_2}$ and $\nu = \frac{x_1}{x_3}$. Chain rule to $F_x$

Let $\kappa = \frac{x_1}{x_2}$ and $\nu = \frac{x_1}{x_3}$. Consider a function $F(\kappa, \nu)$ How to apply chain rule to the following $$\frac{\partial F}{\partial x_1}$$ I am confused that since ...
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0answers
40 views

Functional Derivative with Difficult Chain Rule

I am trying to evaluate the functional derivatives $\dfrac{\delta F[\phi(\textbf{r})]}{\delta n_{1}}$ and $\dfrac{\delta F[\phi(\textbf{r})]}{\delta n_{2}}$ where \begin{gather} F[\phi_1(\textbf{r}), ...

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