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.
1,673 questions
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Step in Brezis' proof of chain rule for Sobolev spaces
I'm going through the proof of Corollary 8.11 in Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations which states:
Let $G \in C^1(\mathbb{R})$ be such that $G(0) = 0$, and ...
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Second Derivative Chain Rule in Matrix Notation
If we have a function $f(x_1,x_2,x_3,x_4)$ and perform a coordinate transformation to $f(y_1,y_2,y_3,y_4)$, then by the chain rule,
$$
\frac{\partial f}{\partial x_1}
=
\begin{bmatrix}\frac{\...
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Confusion in PDE multivariate chain rule
I have a doubt regarding the multivariate chain rule PDE.
Consider an arbitrary function $\phi(x+y+z,x^2+y^2-z^2)=0$. We have to eliminate the function & form a PDE.
The solution as follows:
Let $...
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Chain rule applied to a multivariable function
Consider the function $f(x(s,t),y(s,t))$. Then, the partial derivative with respect to t is defined as:
$\frac{\partial f}{\partial t} = \frac{\partial f}{\partial x}*\frac{\partial x}{\partial t} + \...
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Why, if we drop $f(D_f) \subseteq D_g$ for $f(a) \in D_g$, then chain rule can't hold?
I am having difficulties to formally prove that, in the derivative of composition of two functions $g$ and $f$, the requirement that $f(D_f) \subseteq D_g$ (where $D_f$ and $D_g$ are intervals and ...
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Machine learning: what is the proper name for derivative of a function against a matrix?
In machine learning, it is typical to see a so-called weight matrix. As a low-dimensional example, let this matrix be defined as,
$$W = \begin{bmatrix} w_{11} & w_{12} \\\ w_{21} & w_{22} \end{...
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Calculating a Derivative of a Function
I would like to calculate the derivative of a function. Question has been taken from "Edwards & Penny, Mathematical Analysis and Analytical Geometry" from page 195.
$$ y = (1+2u)^3 , u =...
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The symbol in the chain rule of the partial derivatives
There is a function $z=f(u,\,v),\,u=g(x),\,v=h(x,\,y)$
Can I write the paritial derivative as
$$
\frac{\partial{z}}{\partial{x}}=
\frac{\partial{z}}{\partial{u}}
\frac{d{u}}{d{x}}+
\frac{\partial{z}}{\...
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Compute $\frac{d}{dx} \left( \int_{-x}^x e^{s^2}ds \right)$ and $\frac{d}{dx} \left( \int_{0}^{x^2} \sin(s^2)ds \right)$
From Jiri Lebl "Basic Analysis $1$".
I want to check I am getting these correct, as I have always found applying the second FTOC (fundamental theorem of calculus) difficult, and the book ...
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Chain rule through entropy with respect to distribution's parameters
Define $q_\phi(x)=\int q_\phi(x|z)q(z)dz$ and $\mathcal{H}(q)=-\int q(x)\log q(x) dx$, and suppose that $q(z)=q(\epsilon)=\mathcal{N}(0,I)$ and $q_\phi(x|z)=\mathcal{N}(g_\psi(z), \sigma^2I)$, where $\...
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Solving $(f\circ g)'=f\circ g'$
A student of mine made a silly mistake while taking derivatives:
$$
\color{red}{\partial_x \sqrt{1-x}=\sqrt{\partial_x(1-x)}}=\sqrt{-1}=i.
$$
This has led me to wonder about the problem of finding ...
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Backpropagation chain rule - why is this application valid?
The backpropagation general form, as I have been told, works for any directed acyclic graph. For a proof, I am pointed to the same derivation that was given when there are neat hidden layers, and each ...
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Confusion between these two definitions of the multivariable chain rule
Given a function $f(x, y, z)$, what exactly is the difference between the two following expressions?
$$\frac{df}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{...
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$D(F\circ u)=(F'\circ u)Du$ (chain rule) in Sobolev spaces - still hold when $F'$ discontinuous or when $F$ only weakly diff'ble?
I just finished reviewing my solutions of the exercises in chapter 5 of Partial Differential Equations (2nd Ed) by L.C. Evans. We recall Problem 17:
Exercise 17. (Chain rule.) Let $F\in C^1(\Bbb R)$ ...
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Partial differential equations $ {\partial a(x,y) \over \partial y} = -{\partial u(x,-y) \over \partial y}$
In the following question Prove if a function is holomorphic, then its complex conjugate is holomorphic, by Cauchy-Riemann Equations copper hat writes $ {\partial a(x,y) \over \partial y} = -{\partial ...
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