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Questions tagged [chain-rule]

For questions involving the chain rule in analysis.

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Do specific values of partial derivatives allow us to use the chain rule for functions of 2+ variables?

Given a function $f(x, y)$ and knowing that both x and y are functions of $s$ and $t$, can we use the chain rule to find, for example, the partial derivative of f with respect to s ($\frac{\partial f}{...
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Euler Lagrange equations, chain rule troubles

When considering the first integral the chain rule is used on $$F(y,y’,x)$$ When we do this why do we not consider it as $$F(y(x))$$ As y’ is a function of y But instead as y’ being a separate ...
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22 views

How to evaluate this partial derivative in terms of polar coordinates

How to evaluate this partial derivative in terms of polar coordinates? How to solve this question?
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11 views

Derivate of inverse of composite function

I'm very confused, and this is probably a stupid question. I want to calculate $ \frac{d}{dx} f^{-1}(g^{-1}(x))$. However, I get two seemingly different results taking two different approaches. I. $\...
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1answer
49 views

Is this a version of the chain rule?

From high school maths I know the chain rule as$$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}.$$ So if I wanted to differentiate $y=\cos x^{2}$ I would set $u=x^{2}$ and $y=\cos u$. In this Physics ...
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1answer
26 views

Total differential of a compound function with vector basis

I struggle with this exercice. If $f: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ is differentiable and $g: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ is given by $g(x_1, x_2) = f(x^2_1- x^2_2, 2x_1x_2)$. ...
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2answers
37 views

How does fundamental theorem of calculus and chain rule work?

I came across a problem of fundamental theorem of calculus while studying Integral calculus. A problem: $\frac{d}{dx}\int_{\pi}^{x^2}\cot^2t\ dt$ which was salved as : Step I : Let, F(x) = $\int_{\...
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1answer
56 views

Gradient of trace of squared matrix logarithm

I have a simple question that confuses me for a while: $$f(X) = \text{tr} \left( [ \log(X) ]^2 \right)$$ where $X$ is an $m \times m$ symmetric positive definite (SPD) matrix and $\log(X)$ is ...
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Partial derivative of a multivariate function of a function

I may be missing something super simple here, but it's Friday and my brain can't handle this. I'm looking at the partial derivative of a multivariate function of the form: $\frac{\partial}{\partial ...
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1answer
26 views

About formula of probabilities of entire sequences

In N-Grams section of book "Speech and Language Processing." written by Daniel Jurafsky & James H. Martin found on the Internet, it's said that firstly they represent a sequence of N words either ...
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1answer
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Show that the wave equation takes the form $\frac{\partial^2 u}{\partial r \partial s}$

Show that the wave equation $\frac{\partial^2 u}{\partial t^2} - a^2 \frac{\partial^2 u}{\partial x^2} = 0$ takes the form $\frac{\partial^2 u}{\partial r \partial s} = 0$ under the change of variable ...
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How to apply chain rule to deep functions?

I am working on a deeply nested function. I should find the $\frac{df}{dx}$ considering the following functions: $a = x^2,\ b = \exp(a),\ c = a + b,\ d = \log{c},\ e = \sin{c},\ f = d + e$ My ...
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1answer
69 views

Product rule for trace of matrix functions

I am trying to find the gradient of $f(Z_2) = \|A - Zg(Z_1g(Z_2)) \|_F^2$ with respect to $Z_2$ where $g$ function is applied to each matrix element wise such that $i,j$ element of matrix $g(X) = g(X_{...
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2answers
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$f(3t^3+2,et^2)=(3,6)$ for all $t\in\mathbb{R}$. Prove: $D_f(2,1)$ not invertible

Let $f\in C^1[\mathbb{R}^2 , \mathbb{R}^2]$ satisfying: $f(3t^3+2,e^{t^2})=(3,6)$ for all $t\in\mathbb{R}$. Prove: $D_f(2,1)$ not invertible. My try: we define $g(t)=(3t^3+2,e^{t^2})$. Then, $g\...
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Chain Rule for $\mathbb Q[[T]]$

I am trying to prove the chain rule for formal power series $\mathbb Q[[T]]$, i.e.: Given $f(T) = \sum_{n \geq 0}f_nT^n, \; h(T) = \sum_{n\geq0}h_nT^n$, then: $\frac{d}{dT} f(h(T)) = \left(\frac{d}{...
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1answer
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derivative of scalar function with composition of vector function

Suppose I have a scalar function $f(u(w(v_k))$, where $u=[u_1(w(v_k)),u_2(w(v_k))]$ w is a another scalar function, and $v_k$ is the independent variable. If I was interested in calculating $\frac{...
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Chain rule of C^k function with Sobolev function

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{n}, k \geq 1$ be an integer, and $1 \leq p<\infty .$ Let $u \in W^{k, p}(\Omega) \cap L^{\infty}(\Omega),$ and $\Phi \in C^{k}(\mathbb{R})...
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1answer
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Why is the y-intercept of this calculus problem given like that in the solution?

Given the following problem: Let $f$ be the real-valued function defined by $f(x) = \sqrt{1+6x}$. Determine the slope of the line tangent to the graph of $f$ at $x=4$. Determine the y-...
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Proving the solution of a first order partial differential equation.

I am trying to solve the next partial differential equation, for some region in $\Re^3$: $$ -\frac{\partial \rho}{\partial t}=\frac{\partial \rho}{\partial x} v_{x}+\frac{\partial \rho}{\partial y} ...
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1answer
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Interpretation of composition operator when applying a function to the output of another function

Refreshing my calculus skills a bit, I reviewed the chain rule: I wondered if the composition operation $\circ$ in $g \circ f(x)$ could actually also be written as $g(f(x))$ as this would resemble ...
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Assumption in deriving implicit function theorem?

I'm have a bit of trouble understanding how dependence of variables work with implicit functions. This troubles me during the chain rule and specifically in this case, the implicit function theorem. ...
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Why is that $\int_a^b \frac{\partial f}{\partial x}(x,t)dt = \frac{\partial}{\partial x}\int_a^b f(x,t)dt$

This question is concerned with the integral with parameter, so let's assume that every function below is smooth. To find the formula for the derivative of an integral with parameter, say $$g(x) =...
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Circular working out with partial derivatives

My question is the related to the example below: Why don't we use chain rule when differentiating this: Example: Suppose $x,y$ are functions of $u,v$ and $z = x^2+y$, where \begin{cases} x=e^u \...
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Question regarding proof involving Hermite polynomials. (n:th derivatives)

Theorem: For any $x\in\mathbb{R}$ and $z\in\mathbb{C},$ the hermite polynomials, $$H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2},$$ satisfy $$\sum_{n=0}^\infty H_n(x)\frac{z^n}{n!}=e^{...
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1answer
72 views

Confused about chain rule in Frechet derivative

I've recently started to learn about Frechet derivatives and now have a simple example which I'm not sure if I've solved correctly. To be honest, I've only got a poor understanding of how it works so ...
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1answer
60 views

Basic questions about Frechet derivatives

I had a couple of basic questions about the Frechet derivative. 1) The chain rule: If $F: X\rightarrow Y$ and $G: Y\rightarrow Z$, then $D(G\circ F)(x) = DG(F(x))DF(x)$. The left hand side has a ...
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Chain and product rule for Hadamard product differentiation

(Asked a similar question before but deleted to add further detail) Similar to this question and a related to this question, how can I apply the chain and product rule to find the Jacobian of $$ f_1(...
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What are differential equations and how do you solve ${dy \over dx}=y$ and find $y$ in terms $x$?

I had been wondering about how to solve the equation $${dy \over dx}=y$$. My progress was to use the chain rule, like setting $$z=2x\;{dy\over dx}={dy\over dz}*{dz\over dx}={dy\over d(2x)}*2=y.$$ Now ...
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3answers
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Differentiating $2^{n/100}$ using the chain rule

I am trying to find the derivative of $2^{n/100}$ with respect to $n$. I know that I have to use the chain rule to differentiate this. I have: $$f(n) = 2^{n/100} = e^{\ln2 \cdot n/100}$$ For my $u$...
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Chain rule proof for multivariable functions

I have been looking for a proof similar to the one presented in my book, there are few. But I still can't wrap my head around it. This is how they explain it my book. They begin this by making a ...
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Help finding the derivative of $f(x) = \cos{\left(\sqrt{e^{x^5} \sin{x}}\right)}$

I am trying to find the derivative of $f(x) = \cos(\sqrt{(e^{x^5} \sin(x)})$. I keep getting the wrong answer, and I'm not sure what I'm doing wrong. $$\frac{d}{dx} e^{x^5} = e^{x^5} \cdot 5x^4$$ $...
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1answer
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How to using chain-rule to calculate the gradient in flow chart?

I have an data flow chart as follow The $a$ and $x_1,x_2,x_3$ are vector, $W$ is the matrix Output is the $$y = ((aW+x_1)W +x_2)W+x_3)$$ How to use chain rule to compute $\frac{dy}{dW}$? My ...
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Piecing Together Parts Needed for Multivariable Chain Rule

The Question You want to estimate the rate at which the temperature (degrees F) in a concert hall changes per hour, dT/dt, based upon how many people are in the room, p, and the total heat output of ...
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1answer
68 views

Understanding a step in Rudin's proof of the Inverse Function Theorem

I am reading Rudin's "Principles of Mathematical Analysis", and in a step for the proof on the inverse function theorem he says that the derivative of: $\phi(x)=x+A^{-1}(y-f(x))$ is $\phi'(x)=I-A^{...
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1answer
38 views

Appropriately Applying the Chain Rule to a Substitution made to an ODE

I have the differential equation, $$\frac{dG(-5x)}{dx}=-\frac{1}{6}\left(g_1(x)+\frac{dg_0(x)}{dx}\right).$$ If I make the substitution $u=-5x$, does the above differential equation collapse to form $$...
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Differentiate $\frac{v^4-10v^2\sqrt{v}}{4v^2}$?

I am trying to differentiate $\frac{v^4-10v^2\sqrt{v}}{4v^2}$. I have tried splitting the fraction and doing the division before finding the differential, but I am still not getting the right answer. ...
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1answer
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Stuck in chain rule proof

This is from Apostol's Calculus Vol 2 I have problem with $\lim\limits_{h\to0}\frac{||r(t+h)-r(t)||}{h}$: I think that limit may assume two different values depending on if we approach from left ...
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1answer
35 views

Misunderstanding chain rule in several variables

The question asks what is the wrong reasoning in this statement. If we have $w = f(x,y,z)$ and $z = g(x,y)$ then: $\frac{\partial w}{\partial x} = \frac{\partial w}{\partial x}\frac{\partial x}{\...
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Apply the chain rule with partial derivatives

Hey all, I'm looking for resources on how to apply the chain rule to multi-variate partial derivatives. I'm pretty stumped about how to get ${df\over dx}$ and ${df\over d\mu}$ for the above ...
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Why is continuity of Q(g(x)) needed in this chain rule proof on Wikipedia?

I am looking at the first proof of the chain rule on Wikipedia: https://en.wikipedia.org/wiki/Chain_rule#First_proof In the body of the proof, you will see that the author has defined a function $Q(...
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How much rigour is this proof of multivariable chain rule?

I have seen some statements and proofs of multivariable chain rule in various sites. I "somewhat" grasp them but seems too complicated for me to fully understand them. To make my life easy, I have ...
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1answer
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Sobolev Chain rule in domains with infinite measure

In chapter 4 of "Measure Theory and Fine Properties of Functions" by Evans and Gariepy, part (ii) of theorem 4.4 states: If $f \in W^{1,p}(U)$ and $F \in C^{1}(\mathbb{R})$, $F' \in L^{\infty}(\...
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Chain Rule Second Order PDE

Consider second order PDE: $x\phi_{xx}+2x^2\phi_{xy}-\phi_x=x^3$ My characteristic variables are: $\xi=y-x^2$ and $\eta=y$ $\phi_x = \Phi_\xi.\xi_x+\Phi_\eta.\eta_x=-2x\Phi_\xi$ $\phi_{xx} = -2\...
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Derivatives, chain rules, and inverse functions

Here is a simple question. Let $y=f(x)$ and $z=g(x)$ are two well behaved functions of $x$. How can I calculate the derivative $\frac{dy}{dz}$? Related question is about 2-variable functions $y=f(x_1,...
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Variables change in derivation. What is wrong?

I have two set of variables, which are related: $\left\{ \alpha, \beta, \gamma \right\}$ and $\left\{ v_0, v_1, v_2 \right\} = \left\{ \alpha, \beta, \beta \gamma \right\}$ Now, I want to compute ...
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Find the slope of the tangent line to the curve defined by $7x^4 - 8xy - 6y^3 = 322$ at the point $(2, -3)$

Find the slope of the tangent line to the curve defined by $$ 7x^4 - 8xy - 6y^3 = 322$$ at the pont $(2, -3)$ I'm having a tough time using implicit differentiation and chain rule with all ...
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1answer
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Why does a $\sum_k$ appear when using the chain rule to derive $\delta^L_j?$

I'm following along this book on machine learning. At the moment, the author is proving that \begin{eqnarray} \delta^L_j = \frac{\partial C}{\partial a^L_j} \sigma'(z^L_j) \nonumber\end{eqnarray}...
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1answer
37 views

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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1answer
38 views

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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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 ,...