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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How to express with chain rule in multivariable functions ( higher derivatives)

Since $w=f(x,y)$ , $x=s+t$, $y=s-t$, express $(\frac{\partial^2 w}{\partial x^2} -\frac{\partial^2 w}{\partial y^2})$ in terms of derivatives $\frac{\partial w}{\partial s}$ and $\frac{\partial w}{\...
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Chain rule for composite function with inner function non-smooth

Let $f:\mathbb{R}^2\mapsto\mathbb{R}$ be given by $f(w_1,w_2)=\frac{1}{2}(1-w_2\sigma(w_1))^2$, where $\sigma(x)=\max\{x,0\}$ is the ReLU function. I want to compute the Clarke subdifferential of $f$ ...
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Q in op, f(x)= g(||Ax||^2)-<b,x> , x in R^n [closed]

Let n ∈ N^∗ (n != 0) be an integer, A a real square matrix, and b a vector in R^n. We consider a real function g : R −→ R, g ∈ C^2 (R), and we introduce the function f : Rn −→ R, defined by: f(x)= g(||...
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Derivative of $f(x, y)=x+y$ with respect to $z=(x, y)$

Let $f(x, y) = x + y$ what is the derivative of this function with respect to $z = (x, y)$? I suppose one should use the chain rule, but not sure how it would workout here. Maybe this? $$ \frac{d f(z)}...
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For $\varphi \in C_0^{\infty}$ it holds that $d\varphi (x,v) = D\varphi(x)\cdot v$

Given a test function $\varphi \in C_0^{\infty}$ we have that $d\varphi (x,v) = D\varphi(x)\cdot v$. I read that one has to use the chain rule to prove that but I'm not sure how to incorporate it here?...
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Chain rule for dependent functions

Consider the following equations: $$R=\frac{dU}{dx} +5U\tag{1}$$ $$U=f(x) \tag{2}$$ Which of the following statements is the correct way of calculating the total derivative of R? $$R=R(U(x),x) \...
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Problem in the derivative with multi-index notation

I'm currently having some problems in understanding what our professor wrote. Text reads: $$\partial^{\alpha}\left((x-x_0)^{\alpha} \phi\left(\frac{x-x_0}{\epsilon}\right)\right) = \alpha!\phi\left(\...
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Questions in Proof of Multivariate Chain Rule

I am looking at this proof: https://math.berkeley.edu/~nikhil/courses/121a/chain.pdf and have some confusion. I have seen several proof using little-o notation and am quite confused. Can someone ...
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Extension Faa di Bruno formula

Faa di Bruno's formula gives an expression for the $n$th derivative of a composite function, $\frac{d^n}{dt^n}f(g(t))$, thereby generalising the chain rule. I was wondering whether there is also a ...
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Stuck on a really simple chain rule problem

So I'm going through a PDE book, and working on deriving the fundamental solution of Laplace's equation. The derivation obviously doesn't show all of the calculations/steps involved that are easy, but ...
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Gradient in integral transformation

I have a question on the handling of gradients in coordinate transformations: For a coordinate transformation in form of a rotation defined by the mapping $\Phi: \Omega \mapsto \Omega'$, $\psi \mapsto ...
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Questions Regarding this Chain Rule Proof

I saw this proof of the Chain Rule on Hardy's A Course of Pure Mathematics (the notation I use will be a little different). • Chain Rule: Let $f\,\colon Y \subset \mathbb{R} \to \mathbb{R}$, $g\,\...
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42 views

How do I find $\frac{\mathrm{d}}{\mathrm{d}x} \int_{x}^{\infty} f(t, x) \mathrm{d} t$?

I am trying to compute the following derivative: $$\frac{\mathrm{d}\Big(\int_{x}^{\infty} f(t, x) \mathrm{d} t\Big)}{\mathrm{d}x} \text{.}\tag{1}$$ It is straightforward to compute the following ...
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Theorem 8.5 (The chain rule) from "An introduction to manifolds" by Tu - How should I understand the notation $(...)f$ where $f$ is a smooth function?

Theorem 8.5 If $F : N \rightarrow M$ and $G : M \rightarrow P$ are smooth maps of manifolds and $p \in N$, then $$(G \circ F)_{*, p} = G_{*, F(p)} \circ F_{*, p}$$ Proof Let $X_p \in T_p N$ and $f$ be ...
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Using the chain rule to solve a derivative

I have a known derivative given as: $\frac{dr}{dt} = \frac{a}{r^2}$ And then from that information, I am trying to find: $\frac{d(r^2)}{dt}$ I know that this is equal to $\frac{dr^2}{dr} \times \frac{...
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Application of chain rule in vector functions

a) Compute $\dfrac{\partial f}{\partial \mathbf{x}},$ where $f(z)=\log(1+z)$ and $z=\mathbf{x}^T\mathbf{x}.$ Solution: If $\mathbf{x}=\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix},$ then $\...
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$Z$ has a standard normal distribution. $P(Z \in (-a,b)) = 0.95$ where $a,b > 0$. Find the derivative the length, $l$, of the interval $(-a,b)$.

$Z$ has a standard normal distribution. $P(Z \in (-a,b)) = 0.95$ where $a,b > 0$. I'm asked to find the derivative the length, $l$, of the interval $(-a,b)$. I have an expression for $l$ as a ...
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I have difficulty in this chain rule. Can anyone explain this to me in simple words??

I have difficulty in understanding the chain rule given in the picture. The derivative with $x$ do not have $\lambda$, while $y$ term have. Why is it so???
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When to use Jacobian for Chain rule?

I am following the below theorem and its proof. To replace $\nabla_{\theta}$ to $\nabla_{x}$, it adds $ J_{\theta} G_{\theta}(z) $ using chain rule but I think using gradient $\nabla_{\theta}G_{\...
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A doubt about a proof of chain rule for smooth functions between smooth manifolds

I'm reading Theorem 1.1 in this lecture notes. Theorem 1.1 (Chain Rule for Manifolds). Suppose $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are smooth maps of manifolds. Then: $$ \mathrm{d}(g \circ ...
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Chain rule for smooth functions between smooth manifolds

Following this thread, I'm trying to prove in detail the chain rule to unveil the subtle machinery. Could you have a check on my proof? Let $X \subseteq \mathbb R^M$, $Y \subseteq \mathbb R^N$, and $...
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Directional Derivative for composition of multivariable function

I have been working on "Advanced calculus of several variables" by C. H. Edwards, and I saw question 3.3 (c) on page p.88. 3.3(c) says that $F : \Bbb R^n \to \Bbb R^n$ differentiable ...
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Chain rule with$~\left(\frac{1}{2}\ln\left(\frac{x^{2}+\sqrt{2}x+1}{x^2-\sqrt{2}x+1}\right)+\tan^{-1}\left(\frac{\sqrt{2}x}{1-x^{2}} \right)\right)$?

Please jump to the bottom of this post if you just want to see my current critical problem. $$y=\frac{1}{2\sqrt{2}}\left(\frac{1}{2}\ln\left(\frac{x^{2}+\sqrt{2}x+1}{x^{2}-\sqrt{2}x+1}\right)+\tan^{-1}...
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Why the results from chain rule and partial derivative do not match?

Given $f(x,y) = xy^2/(x^2+y^2)$, if $x^2 + y^2 > 0$ else $f(x,y) = 0$ where $x(t)=t, y(t)=t$ If we apply chain rule: $f(t) = t^3/(2t^2) = t/2$ then $f'(t)=1/2$ If we try to calculate partial ...
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How to apply Chain Rule with differentials in Matrix Derivatives?

@Steph had kindly answered my other question, but I can't work out the math. He said that "The correct way to apply chain rule with matrices is to use differentials", and provided the answer ...
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Why the order of Chain Rule all weird in matrix derivatives behind machine learning?

Suppose I have a neural network as the image. And to make it simple, I'll set all the activation functions to $f(x)=x$. So if we set the nodes as Row Vectors, $A_1$ is a $1 \times 2$ matrix, $W_1:2 \...
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Chain rule for higher dimension question

My physics textbook states that $$\frac{\partial f(y+\alpha \eta, y' + \alpha \eta', x)}{\partial \alpha} = \eta \frac{\partial f}{\partial y} + \eta' \frac{\partial f}{\partial y'}.$$ I'm trying to ...
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Chain rule and derivative with matrix product?

I'm trying to compute some derivatives with given vectors and functions: column vector $X=[x_1,x_2,\dots,x_n ]^T$ and $Z=[z_1,z_2,\dots,z_n ]^T$, row vector $Y=[y_1,y_2,\dots,y_n ]$ $f(X,Y)=e^{XY}$ $$...
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Chain rule with exponential map

Let $G$ be a Lie group, $X,Y \in $ Lie($G$) and $f \in C^{\infty}(G)$. Then, by the chain rule: $\frac{d}{dt} \Bigr \lvert_{t=0} f(exp(tX)exp(tY)) = \frac{d}{dt} \Bigr \lvert_{t=0} f(exp(tX)exp(0 \...
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simple q, partial derivative writing formally with function having both implicit and explicit dependence on variable

I have the function $L$ which depends on other variables as well as $n$ explicitly and implicitly as below. $$\frac{\partial L}{\partial n} = \frac{\partial L}{\partial e}\frac{\partial e}{\partial n} ...
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Chain Rule Puzzle

Here is an old question: If $\boldsymbol{s_r}$ denote the sum of the $\boldsymbol{r}$ th powers of the roots of the equation $$\boldsymbol{x^n+p_1x^{n-1}+\cdots +p_n=0}$$ prove that if the ...
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understanding a step in the proof of the chain rule

statement of the chain rule: let $f: A \to R$ and $g: B \to R$. s.t. $f(A) \subset B$ where $ g ∘ f$ is defined. If $f$ is differentiable at $a \in A$ and $g$ is differentiable at $f(a) \in B $, then ...
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How can I use the chain rule in this strange case? [closed]

I have to do the partial derivative of a compound function $Z$ with respect to $X$ and then with respect to $Y$. I have that $X$ is my independent variable, then $Y$ depends on $X$, and $Z$ depends ...
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1answer
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Computing gradients with chain rule

Let $x_{1}, \dots, x_{N}$ be a sequence of vectors in $\mathbb{R}^{n}$ and $A$ be an $n \times n$ matrix. Let $f: \mathbb{R} \to \mathbb{R}$ be a smooth (as smooth as you want) function. We define $$ ...
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Chain rule substitution differentials

I'm reading through Bamberg and Sternberg, and I'm on Chapter 5. It has the attached passage. I understand it up until it says we can substitute our $dy=15x^2dx$ into the $d(y^2)=2y\space dy$ equation....
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Why can't I use substitution to derive the probability density function of the log-normal distribution?

I can see that almost the same question was asked here, but I didn't really understand the answers, so I was wondering if someone could help me get a better grip on it. Here is the proof I came up ...
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Simplified estimate for derivatives of $f\circ g$?

Let $g:\mathbb R\to \mathbb R$ be a smooth compactly supported function, and let $f:\mathbb R\to \mathbb R$ be a smooth function with $f(0)=0$. Let $\|f\|_m := \|f\|_{C^m}=\sum_{0\le k\le m}\sup_{x\in\...
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Chain rule for change of variables (2nd order pde)

Let $x,y$ be independent variables, and we transform into $\xi = \xi(x,y), \eta = \eta(x,y)$. By the chain rule, we clearly have: $$ \frac{\partial u}{\partial x}=\frac{\partial u}{\partial \xi} \frac{...
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find derivative of $\sqrt{x}+\sqrt{y}=\sqrt{a}$

I have an Implicit Function $\sqrt{x}+\sqrt{y}=\sqrt{a}$ the graph of the function is I need to prove that $p+q=a$ and I need to find $\frac{d}{dx}$ to find the the slop to prove that. result: ...
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Correct way to apply chain rule

Let $f(x,y) = x + x^2 + y$, we have $f_x(x,y) = 1 + 2x$. Let $u = x^2$ and write $f(x,y,u) = x+ u +y$. When applying the chain rule, $f_x(x,y,u) = f_u(x,y,u)(du/dx) + f_y(x,y,u) (dy/dx) + f_x(x,y,u) (...
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Find second partial derivative of G by x where $G(x,y) = F(x,y,f(x,y))$

Let $G(x,y)=F(x,y,f(x,y))$. I want to find $\dfrac{\partial^2G}{\partial x^2}$ in terms of partial derivative of $F$ and $f$. I am not sure if my solution is correct as it looks different from the one ...
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What is the advantage of using Gradian to measure an angle?

What is the advantage of Gradian to measure an angle? For example, I know radian is useful in Calculus because e.g. it simplifies the derivative of trigonometric functions. By the way, except the ...
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How to derive a summation if the sum is constrained by a function of the variable I'm deriving with respect to

Imagine we have a function $f(x,y)$ that can be expressed as \begin{equation} f(x,y)= \sum_{k=0\\ k\notin \mathcal{K}}^\infty e^{-ikx}\bar{f}(k,y) \end{equation} where the sum goes over all natural ...
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If $ f(\theta) $ is differentiable and $ g(x) = f(\theta + ax) $, can we show that $ g'(0) = a \frac{df}{d\theta} $?

Problem Let $ f(\theta) $ be a differentiable function. Let $ g(x) = f(\theta + ax) $. Can we show that $ g'(0) = \left[ \frac{dg}{dx} \right]_{x=0} = a \frac{df}{d\theta} $? Example It appears to be ...
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How does the chain rule imply that for a solution to the diffusion equation, we have an invariance that $u(x,t)=u(1-x,t)$?

Consider the diffusion equation $u_{t}=u_{x x}$ in $\{0<x<1,0<t<\infty\}$ with $u(0, t)=u(1, t)=0$ and $u(x, 0)=4 x(1-x) .$ $\text {Show that } u(x, t)=u(1-x, t) \text { for all } t \geq 0 ...
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How to show using the chain rule that $u_{x x}+u_{y y}=u_{r r}+\frac{1}{r} u_{r}+\frac{1}{r^{2}} u_{\theta \theta}$? [closed]

In solving a potential problem by the separation of variables method in a circula it is necessary to express the problem in polar coordinates. By setting $$ x=r \cos \theta, \quad y=r \sin \theta $$ ...
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117 views

Simple use of the chain rule

I have the following problem where I feel I am missing something obvious: I have the function $f(x)=-\ln(x+x^2)$ with $x>0$ and I wish to find the derivative. By using the chain rule I get $$ f'(x)=...
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How can I compute the derivative of a function of 't' mapped to another function that has a function within it but no actual 't' variables?

I won't give the exact example but let's say I have this function $G: t ↦ F(Y(t))$ I know this to translate to $G(t) = F(Y(t))$ If $F(y)$ is some complicated equation with constants $a, b$, such as $y^...
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38 views

Proving that division in the multivariable chain rule is "allowed"

I have always had an issue with how one seems to be able to multiply through by say $dx$, which shouldn't be really allowed as $dy/dx$ is really $d/dx$ operating on y. I would like to rigioursly prove ...
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Multivariable Chain Rule for Implicit Multivariable Functions?

I'd like to compute $\frac{\partial x}{\partial z}$ along $S$ at $(x,y,z)$ for $S: \frac{1}{x}+\arctan(y+2z)=1$. My Approach: I can define $w(x,y,z)=\frac{1}{x}+\arctan(y+2z)$ and find the total ...

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