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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Chain Rule Intuition
We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...
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Understanding the chain rule in probability theory
When my teacher told us about the chain rule I found it quite easy, but when I am trying to prove something based on this rule I kind of get confused about what are the allowed forms of this rule. For ...
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What's my confusion with the chain rule? (Differentiating $x^x$)
When deriving $x^x$, why can't you choose $u$ to be $x$, and find $\dfrac{d(x^u)}{du} \dfrac{du}{dx} = x^x$? Or you could go the other way and find $\dfrac{d(u^x)}{du}\dfrac{du}{dx}$, giving $\ln(x)\...
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Why can you mix Partial Derivatives with Ordinary Derivatives in the Chain Rule?
This question is a simplified version of this previous question asked by myself.
The following is a short extract from a book I am reading:
If $u=(x^2+2y)^2 + 4$...
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Why is this famous proof of the chain rule called "technically incorrect" in this pdf? [duplicate]
So I was looking through various proofs of the chain rule...and I came across this paper. The first proof given is complete and quite well-explained. But another simplistic proof is given in the end......
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Why is there only one term on the RHS of this chain rule with partial derivatives?
I know that if $u=u(s,t)$ and $s=s(x,y)$ and $t=t(x,y)$ then the chain rule is $$\begin{align}\color{blue}{\fbox{$\frac{\partial u}{\partial x}=\frac{\partial u}{\partial s}\times \frac{\partial s}{\...
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Derivative of an even function is odd and vice versa
This is the question: "Show that the derivative of an even function is odd and that the derivative of an odd function is even.
(Write the equation that says $f$ is even, and differentiate both ...
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Derivation of the multivariate chain rule
I can't believe I couldn't find this information online, but could someone provide me a good proof of the multivariate chain rule ?
\begin{align}
\frac{df}{dt} = \frac{df}{dx}\frac{dx}{dt} + \frac{df}{...
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When is the derivative of $f(g(x))$ equal to $g(f'(x))$?
By the chain rule, we know that the derivative of $f(g(x))$ is $f'(g(x))g'(x)$.
Question: When is the derivative of $f(g(x))$ equal to $g(f'(x))$?
Trivial solutions include the following:
Let $f$ ...
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About a chain rule for Wronskians
The Wronskian of $(n-1)$ times differentiable functions $f_1, \ldots, f_n$ is defined as the determinant
$$
W(f_1, \ldots, f_n)(x) = \begin{vmatrix}
f_1(x) & f_2(x) & \cdots & f_n(x) \\
...
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Understanding the chain rule in the Wirtinger calculus
The Wirtinger differential operators are defined by:
\begin{equation}
\frac{\partial}{\partial z} = \frac{1}{2}\left(\frac{\partial}{\partial x} - i\frac{\partial}{\partial y}\right) \\
\frac{\...
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Integrating composite functions by a general formula?
There is a "Chain - Rule" in calculus that allows us to differentiate a composite function in the following way (Sorry for mixing up the two standard derivative notations.) :
$$\frac {d}{dx} f(g(x)) =...
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Neural Networks and the Chain Rule
With neural networks, back-propagation is an implementation of the chain rule. However, the chain rule is only applicable for differentiable functions. With non-differentiable functions, there is no ...
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Chain rule characterizes the derivative?
To what extend does the chain rule $(f\circ g)'=(f'\circ g)\cdot g'$ characterize the derivative $(\ldots)'$?
More precisely:
If a non-constant operator $T$ defined on all differentiable functions, ...
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Differentiation Chain Rule for Quaternion Functions of Quaternions
I am trying to calculate the time derivative of a quaternion function:
$$\mathbf{p}\left(t\right)=\left(\mathbf{q}\left(t\right)\right)^{\tau}$$
Where I've used bold font to indicate that $\mathbf{p}...
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Chain rule proof - Apostol
Apostol calculus I page 174-175 has the proof of chain rule.
Theorem states: Let f be the composition of two functions u and v, say $f=u \circ v$. Suppose that both derivatives $v'(x)$ and $u'(y)$ ...
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Spivak's Chain Rule Proof (Image of proof provided)
If $g$ is differentiable at $a$, and $f$ is differentiable at $g(a)$, then $f \circ g$ is differentiable at $a$, and
$$
(f \circ g)^{\prime}(a)=f^{\prime}(g(a)) \cdot g^{\prime}(a).
$$
Define a ...
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Derivative of norm 2
I'm struggling a bit using the chain rule.
Given the function $\phi$ defined as:
$\phi(x) = ||{A\bf{x}-b}||_2$
where $A$ is a matrix and $b$ is a vector.
What is the gradient $\nabla\phi$ and how ...
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On the 'wrong proof' of the chain rule
I am looking through an old analysis course that I had and I was pondering a bit about the proof of chain rule (especially the notorious wrong proof that you can give). I'd be happy if someone was ...
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Why is the "correct" proof of the chain rule correct? What is actually happening here?
There is a correct and an incorrect proof going around when it comes to the Chain Rule (see below). The problem with the incorrect proof is that $g(x)-g(a)$ might be $0$ if $x\to a$ creating a ...
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Why is the following NOT a proof of The Chain Rule?
In Leibniz notation of the chain rule,
$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$
Where $y\left ( u\left ( x \right ) \right )$ is a composite function of x.
I understand that the du's ...
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Why is this proof of the chain rule incorrect?
I saw this proof of the chain rule but it says this is a flawed proof. Why? I guessed the reason it is wrong because you can't substitute $g(x+h)$ and $g(x)$ into in limit.
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How much rigour is this proof of multivariable chain rule?
I have seen some statements and proofs of multivariable chain rule in various sites. I "somewhat" grasp them but seems too complicated for me to fully understand them.
To make my life easy, ...
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Why is it wrong to derive the chain rule this way?
My book says that the chain rule can stated as $$\dfrac{dy}{dx} = \dfrac{dy}{dt} \dfrac{dt}{dx}$$
However, it the book says that it is incorrect to reason that the chain rule is true because the $dt$'...
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Does the converse of the chain rule hold in general?
I've found the following theorem for single variable real valued functions:
The converse of the chain rule: Suppose that $f,g$ and $u$ are related so that $f(x)=g(u(x))$. If $u$ is continuous at $...
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Using chain rule to differentiate $f(x)=a(x)b(x)$?
Why can I not apply the chain rule to a product in the following way.
If we have some product:
$$f(x)=a(x)b(x)$$
Consider the multiplication of b by a as another’s function so that:
$$f(b(x))=ab$$
...
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A puzzling "failure" of the Chain Rule
Consider the standard transformation equations between Cartesian and polar
coordinates:
\begin{align*}
x&=r \cos \theta\\
y&=r \sin \theta
\end{align*}
and the inverse: $r=\sqrt{x^2+y^2}, \...
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Taking the derivative of $y = (\frac{x}{1-\sqrt{x}})^3$ using the chain rule
While working through differential calculus questions for the chain rule, I stumbled upon:
$$y = \left(\frac{x}{1-\sqrt{x}}\right)^3 $$
I initially attempted to apply the chain rule, but to apply ...
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Does there exist a gradient chain rule for this case?
My question comes from this article in Wikipedia. I noticed that there is a chain rule defined for the composition of $f:\mathbb{R}\to\mathbb{R}$ and $ g: \mathbb{R}^n \to \mathbb{R}$ given by
$$
\...
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How to add the derivative of a matrix to the chain rule?
In machine learning, I'm optimizing a parameter matrix $W$.
The loss function is$$L=f(y),$$where $L$ is a scalar, $y=Wx$, $x\in \mathbb{R}^n$, $y\in \mathbb{R}^m$ and the order of $W$ is $m\times n$.
...
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Proof of the chainrule: is this proof correct and did I use the right notation?
I created this proof of the chainrule. Being a (relative) beginner at math I have a few questions.
Is the proof below correct? I was especially in doubt about the use of $h$ on both sides.
Is the (...
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"Binomial" partial derivatives/PDE
I have encountered the following expression,
$$x^2 \frac{\partial^2 u}{\partial x^2} + 2 x y \frac{\partial^2 u}{\partial x \partial y} + y^2 \frac{\partial^2 u}{\partial y^2}=C,\tag{1}$$
where $u=f(x,...
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How to simplify $\frac{\partial^m}{\partial y_i^m}\mathrm{div }(A\nabla u({\bf x}({\bf y})))$
Let $\bf{x}\in\mathbb{R}^n$ (interesting in $n\in\{2,3\}$) and let $A=A_{n\times n}=\mathrm{diag}(a_1({\bf x}),\dots,a_n({\bf x}))$, that is
$$A_{2\times2}=\begin{bmatrix}
a_1(\bf{x})&0\\0&...
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Simplified estimate for derivatives of $f\circ g$?
Let $g:\mathbb R\to \mathbb R$ be a smooth compactly supported function, and let $f:\mathbb R\to \mathbb R$ be a smooth function with $f(0)=0$. Let $\|f\|_m := \|f\|_{C^m}=\sum_{0\le k\le m}\sup_{x\in\...
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Spivak Chapter 10, Exercise 19 solution verification. Prove that if $f^{(n)}(g(a))$ and $g^{(n)}(a)$ both exist, then $(f\circ g)^{(n)}(a)$ exists.
This question is asked here and here.
I'm going to try to answer it for my own edification, and for the next poor soul that comes across it and has trouble :)
I'm using Spivak's Answer Book solution, ...
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Multivariate Chain Rule and second order partials
For the function
$g(t) = f(x(t),y(t))$, how would I find $g''(t)$ in terms of the first and second order partial derivatives of $x,y,f$? I'm stuck with the chain rule and the only part I can do is:
$$...
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Proof of multivariable chain rule
I'm working with a proof of the multivariable chain rule $\displaystyle{\frac{d}{dt}g(t)=\frac{df}{dx_1}\frac{dx_1}{dt}+\frac{df}{dx_2}\frac{dx_2}{dt}}$ for $g(t)=f(x_1(t),x_2(t))$, but I have a hard ...
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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 ...
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Simpler proof that $y^3[d^2y/dx^2]$ is a constant if $y^2=ax^2+bx+c$?
here's my question
If $y^2=ax^2+bx+c$ then prove that $y^3[d^2y/dx^2]$ is a constant .
I have solved this using the conventional method, taking square root, differentiating w.r.t to x and using ...
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Multivariable Chain Rule Formula doesn't make sense to me
Consider $z=f(x(t),y(t))$, then its chain rule formula is:
$\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}$
What I cannot make sense of, are ...
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$n$th derivative of $\sin(f(x))$
This problem came up in some math that I am working out on my own, not from a textbook, so there may not be any solution.
$$g(x) = \sin(f(x))$$
For any polynomial function $f(x)$,
$$g'(x)=\cos(f(x))...
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Help finding the derivative of $f(x) = \cos{\left(\sqrt{e^{x^5} \sin{x}}\right)}$
I am trying to find the derivative of $f(x) = \cos(\sqrt{(e^{x^5} \sin(x)})$.
I keep getting the wrong answer, and I'm not sure what I'm doing wrong.
$$\frac{d}{dx} e^{x^5} = e^{x^5} \cdot 5x^4$$
$...
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Differentiate $\arcsin(\frac{1}{\sqrt{1 + x^2}})$
This is from Calculus by Michael Spivak, 3rd Edition, Chapter 15, problem 1 (iv):
Differentiate $f(x) = \arcsin\left(\frac{1}{\sqrt{1 + x^2}}\right).$
Here's my attempt:
\begin{align}f'(x) &= \...
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Using chain rule to find a partial derivative
I'm currently studying for the math GRE and this question was on one of the recent practice exams. I will restate the full question below, then provide my solution.
Let $u(x,y)$ and $v(x,y)$ be real-...
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Applying the chain rule to a function mapping matrices to matrices
Given a function $F:\text{Mat}(n,n)\rightarrow\text{Mat}(n,n)$, I'd like to get $\frac{d}{dM_{ij}}F(M)^2$ using matrix calculus.
I already derived it by simply writing it out explicitly for my choice ...
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Is it possible to differentiate $\sin x$ with respect to $\cos x$ from first principles?
I was doing a practice problem today for a University admissions test, where it asked me to differentiate $\sin x$ with respect to $\cos x$. The solution I found used the chain rule:
\begin{align}
\...
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Special case of chain rule
Suppose $H$ is a Hilbert space, $I:H \to \mathbb{R}$ is a functional and $\eta_t:\mathbb{R} \to H$.
I want to understand why
\begin{equation*}
\frac{d}{d t} I\left(\eta_t\right)=\left(I^\prime\left[\...
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Partial Derivative Chain Rule When Variables Are Not Independent
Let's say, $x$ is a function of $t$ ($x = x(t)$) and $y$ is a function of $t$ ($y = y(t)$). And, $f$ is a function of $x$ and $y$ ($f = f(x, y)$). Then by the chain rule $$\frac{df}{dt} = \frac{\...
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Is it possible to solve $f'(x)=f^{-1}(x)$?
I am interested in proving what family of functions have the property
$$f'(x)=f^{-1}(x)$$
I've never dealt with a differential equation of this form, hence I could only go as far as to gather a little ...
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Extracting diagonal of $J^TJ$ via automatic differentiation like techniques
First of all please, let me know if this question is more suited for scicomp.stackexchange.com or or.stackexchange.com, and ...
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