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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Given $\theta = \arccos(5x^3)$, finding $d\theta/dx$ implicitly and with the chain rule
I am having trouble with a question on my homework and I have done some work but I'm not sure if I have completed all that is asked or how to take it further if needed.
Let $\theta = \arccos(5x^3)$. ...
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How can I use he following similarity transformation in this pde to get to the equation in the image? [closed]
I don't know how to get from equation 1.20 to equation 1.21
I tried to use the chain rule for partial derivatives. If $f = f(x,t)$, I wrote:
$\frac{\partial f}{\partial t} = \frac{\partial f}{\...
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Derivative of $x^x$: why wrong answer? [duplicate]
I know by taking log on both sides, one can derive the derivative of $x^x$, which is $x^x \left(\ln x+1\right)$. I tried another method but got the wrong answer. I regarded the base $x$ as another ...
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Derivative of trace of kronecker multiplied with another matrix
I have the expression
$$
f(\mathbf{X}) = \text{tr}(\mathbf{G} (\mathbf{A} \otimes \mathbf{X}))
$$
and I am looking for the derivative with respect to $\mathbf{X}$.
Some additional information: $\...
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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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Correct Application of Chain Rule?
\begin{align*}
&\frac{d}{dx} 4(2x + \sqrt{x^3+3})^{-2}\\\\
=\text{ }&4 \cdot \frac{d}{dx} (2x + \sqrt{x^3+3})^{-2}\\\\
=\text{ }&4\frac{-2}{(2x+\sqrt{x^3+3})^3} \cdot \frac{d}{dx}(2x + \...
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Does $y\to x,y’’\to-\frac{y’’}{y’^3}$ always convert a differential equation into that of the inverse function’s?
The inverse substitution $y\to x$ implies $\frac{dy^{-1}(x)}{dy}=\frac1{y’(x)},\frac{d^2y^{-1}(x)}{dy^2}=-\frac{y’’(x)}{y’^3(x)}$, so $y’\to \frac1{y’}$ and $y’’\to-\frac{y’’}{y’^3}$.
It seems to ...
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Formalising a specific coordinate transformation
Suppose I have a PDE given as follows,
$$\left\{\frac{\partial^2}{\partial r^2} + \left(\frac{2}{r} + \frac{f'}{f}\right)\frac{\partial}{\partial r} - \frac{1}{f^2}\frac{\partial^2}{\partial t^2}\...
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Second derivative calculation
Consider $\boldsymbol{\psi}(\boldsymbol{q})\in R^2$ is a function of vector $\boldsymbol{q}(t)\in R^3$ which is a function of time $t\in R^+$. The first time derivative of function $\boldsymbol{\psi}$ ...
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Simple derivative computation for time dependent function
Let $F:\mathbb{R}^m\to \mathbb{R}$ be a smooth function. Consider the function $G:\mathbb{R}^m\times \mathbb{R}^m\times (0,1)\to \mathbb{R}$ defined by $$G(u,v,t)=F\left(\dfrac{u-v}{t}\right)$$
...
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Problem 3-40 in "Calculus on Manifolds" by Spivak. Show that we can write $g=g_n\circ\cdots\circ g_1$ if and only if $g'(x)$ is a diagonal matrix.
I am reading "Calculus on Manifolds" by Michael Spivak.
Problem 3-40. If $g:\mathbb{R}^n\to\mathbb{R}^n$ and $\det g'(x)\neq 0$, prove that in some open set containing $x$ we can write $g=T\...
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Chain rule applied on Jacobian
Say we want to find the acceleration vector in spherical coordinates and in cartesian basis.
By defining the position vectors in cartesian, $\mathbf{x}=(x,y,z)$, and in spherical coordinates, $\mathbf{...
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Theorem 28.4 [Chain Rule] Ross Elementary Analysis
The following is a segment of Ross's proof of the chain rule.
$f$ is defined on some open interval $J$ containing $a$. $f$ is differentiable at $a$ and suppose $f(x) = f(a)$ for $x$ arbitrarily close ...
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Application of the Chain Rule in the Proof of Hadamard's Lemma
I read the proof for Hadamard's lemma available on Wikipedia but I do not understand the application of the chain rule in the proof.
In particular, $f$ is a smooth function defined on an open star-...
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Chain rule when mixed partial multiplied by first order partial
Given
$$
r = \frac{{\partial s}}{{\partial t}}
$$
If we take the partial derivative of $r$ with respect to $u$ $\frac{{\partial r}}{{\partial u}}$ and multiply it by $\frac{{\partial u}}{{\partial v}}$...
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Sketchy use of multivariable chain rule under too weak hypoteses
I've found this statement in my real analysis course notes:
Let $f: B_r (x_0) \subseteq \mathbb{R}^m \to \mathbb{R}$ ($B_r (x_0) = \{ x \in \mathbb{R}^m : d(x, x_0) < r \}$) be a function such ...
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Calculus implicit differentiation question
I stumbled upon this calculus implicit differential question: find
$\cfrac{du}{dy} $ of the function $ u = \sin(y^2+u)$.
The answer is $ \cfrac{2y\cos(y^2+u)}{1−\cos(y^2+u)} $. I understand how to ...
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Differentiation- proof by Induction
Here is my problem:
"Suppose f is a differentiable function whose domain is $(-\infty,\infty)$. We define an infinite sequence of functions $f_n(x)$ as follows:
$f_1(x)=f(x), f_2(x)=f(f_1(x))$, ...
CommunityBot
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Multivariate Chain Rule with Derivatives in Intermediate Functions
I have a function
$$G: \mathbb R^d \times \mathbb R \times \mathbb R^d \to \mathbb R$$
where $d$ is a positive integer and the arguments of $G$ are denoted by $({\bf y}, z, {\bf p})$. I'm denoting ...
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Using the Chain Rule to Switch Between Coordinate Axes in a PDE
I have a system of PDEs that are written in terms of two coordinates: $(x,z)$ which are the usual Cartesian coordinates and $(l,n)$ which are tangential and normal components to some deformable ...
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Application of Chain Rule in Proof of Stokes' Theorem
In the proof of Stokes' theorem in "Vector Calculus" by Marsden and Tromba, I noticed that the chain rule is applied selectively. Specifically, the chain rule is not used when expanding the ...
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Chain rule to differentiate vector fields with a connection or Lie derivative
Let $X : M \rightarrow TM$ be a smooth vector field, $f : M \rightarrow \mathbb{R}$ a smooth real valued function and $\gamma : [0,T] \rightarrow M$ a smooth curve in $M$.
I am interested in ...
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Why does the transformation of a random variable require the transformation have continuous derivative and the original pdf be continuous?
In Statistical Inference (2e, Casella & Berger), there is the following theorem (p. 51) (emphasis mine):
Theorem 2.1.5 Let $X$ have pdf $f_X(x)$ and let $Y = g(X)$, where $g$ is a monotone ...
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Applying the Chain Rule for ($\sin(x) + 1$)
So, according to the chain rule,
$$
\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
$$
In this case, shouldn't ...
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Multivariable Chain Rule with a scaled Function
Let $f: \mathbb{R}^d \setminus \{0\} \to \mathbb{R}$ be continuously differentiable. For $x \in \mathbb{R}^d\setminus \{0\}$ and $\lambda \in (0,\infty)$, define the function $g_x : (0,\infty) \to \...
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Double derivative representation in terms of $\theta$?
Ok, this is a thought I had, as we all know that we can write $\mathrm dy/\mathrm dx$ as $\tan(\theta)$ because it's just the slope of the tangent line. Thought I could get the double derivative in ...
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Instance of chain rule in Spivak's Calculus on Manifolds after implicit function theorem [duplicate]
I have a question about a use of the chain rule in Spivak's Calculus on Manifolds. I believe this is just some basic confusion I have with the notation.
Let $f:\mathbb R^n\times\mathbb R^m\to\mathbb R^...
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Rewriting the second derivative of a function by substitution
I would like to know if the equation
$$
\frac{d^2T(x)}{dx^2} = \frac{1}{2}\cdot\frac{d}{dT}\left(\frac{d}{dx}T(x)\right)^2\quad(1)
$$
is true for a general function T(x). The function T(x) describes ...
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Chain rule for partial derivatives
Minton and Smith, in "Calculus" define the chain rule for full derivatives $\frac {dz} {dt}$ as it follows:
$$\bbox[5px, border: 1.5px solid olive]{{\color{olive}{\text{THEOREM }1.1}~(\text{...
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Calculate mixed second partial derivative
Given a function $f: \mathbb{R}^2 \to \mathbb{R}$ with continuous partial derivatives. It is given in addition that:
[ f'x (3,9) = f'y (3,9) = f''{xx} (3,9) = f''{yy} (3,9) = 1 ]
Define $g(x, y) = f(x^...
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Chain rule with functional derivatives?
I'd like to make the functional derivative of the functional $S[\phi(x)]$ with respect to the Fourier transform $\widetilde{\phi}(p)$ such that
$$\phi(x)=\int\frac{d^{d}p}{(2\pi)^{d/2}}e^{ip\cdot x}\...
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Multivariable chain rule for function with one negative component
I have a (smooth) function $\psi:\mathbb{R}^2 \to \mathbb{R}\: : \: (y_1,y_2) \mapsto \psi(y_1,y_2)$. Now I want to calculate the partialderivative of $\psi(-y_1,y_2)$ with respect to $y_1$. I thought ...
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Wirtinger Matrix Derivative Chain rule
I'm trying to compute the matrix Wirtinger derivative
$$\frac{\partial (f\circ g)(Z)}{\partial Z}$$
where $g(Z) := B(A Z-Z A)$
and $f(g(Z)):= \mathrm{Tr}\left(\sqrt{g(Z)^* g(Z)}\right)$.
Here $Z$ is a ...
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Chain rule mismatch
Let
$$
g\bigl(h(t)\bigr) = \cos(\sqrt{t})
$$ and
$$
g(t) = \cos(t)
$$
and
$$
h(t) = \sqrt{t}.
$$
Verify that
$$
\frac{dg}{dh} = \frac{\;\frac{dg}{dt}\;}{\frac{dh}{dt}}.
$$
I tried doing this but I ...
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Why is the numerator-layout Jacobian transposed in backpropagation calculation?
In the derivation of the backpropagation algorithm in Neural Network Design by Hagan et al., we consider the derivative of the scalar-valued sample loss function $\hat{F}$ with respect to a vector of &...
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Counter example: if each function of a composite function being not differentiable at a point, then the composite function is also not differentiable
I was reading counter example in calculus book. And stuck one problem:
If a function $g(x)$ is not differentiable at $x=a$ a and a function
$f(x)$ is not differentiable at $g(a)$, then the function $...
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Chain rule for Clarke-derivatives
The Clarke-gradient is often introduced to extend ideas from convex analysis to non-convex functions, see [Clarke, Sec 2.1]. In particular, given $f:\mathbb{R}^n\rightarrow \mathbb{R}$ Lipschitz in $x$...
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About the chain rule of the exponential entropy
In the paper unifying framework of information measures the conditional exponential entropy (see equation 29) is defined as:
$\mathcal{E}_{\alpha}(X|Y) = E_y\left(\int_{\mathbb{R}} f^{\alpha}(x|y)\,...
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Applying chain rule/product rule to $\frac{d}{d\tau} \left( \frac{d}{dt} \theta(t(\tau)) \, \frac{d}{d\tau} t(\tau) \right)$
How can I apply the chain rule to following function?
$$
\frac{d}{d\tau} \left( \frac{d}{dt} \theta(t(\tau)) \, \frac{d}{d\tau} t(\tau) \right)
$$
The right way to rewrite the above expression is by ...
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Can $F(x)=g(f(x))$ be differentiable at $x=\alpha$ if f and g are not at $x=\alpha$
is $F(x)=g(f(x))$ always non-differentiable at $x=\alpha$ if:
a) f is differentiable at $\alpha$ and g is not différentiable at $f(\alpha)$
b) f is not différentiable at $\alpha$ and g is ...
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Derivative of an even function is odd and vice versa
This is the question: "Show that the derivative of an even function is odd and that the derivative of an odd function is even.
(Write the equation that says $f$ is even, and differentiate both ...
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Generalized chain rule for multivariable functions
Let x(t) = $(x_1(t),\dots,x_m(t))$ and let $f: \mathbb{R}^m \to \mathbb{R}$ be a multivariable function. Then let $\hat f$(t) = f (x(t)).
I know that $$\frac{d\hat f(t)}{dt} = \sum_{h=1}^m \frac{\...
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How do I calculate the derivative of a composition $R^{n} \rightarrow R^{n \times n} \rightarrow R^{n}$?
I am having problems calculating the derivative of a function.
Let $C:\mathbb{R}^{n \times n} \longrightarrow \mathbb{R}^{n}$ with $C(M) = (I - M)^{-1}(I + M)x_0$ for (I - M)
invertible ($x_0 \in \...
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Finding a derivative for option pricing
I am trying to find the derivative of the expression below wrt $S$:
$$
e^{-dt}N(x)
$$
where $x$ is defined as:
$$
\frac{\ln({\frac{S}{K}})}{\sigma\sqrt{t}}
$$
and $N(x)$ is the cumulative of the ...
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Chain rule with a variable held constant (or along a constant surface) [duplicate]
Say we have a Cartesian function, $f(x,y)$ and we move to a polar coordinate scheme
$g(r,\theta)$,
$$g(r,\theta) = f(x,y) \\ \therefore x = x(r,\theta), y = y(r,\theta)$$
Just to lay out the ...
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Strange usage of chain rule. Can anyone explain why this derivation was done this way?
There are 2 issues I have with the way this was done. The first was how chain rule was used in the (1.35), and the second was how chain rule was used in (1.36). It all seems so counterintuitive.
For (...
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Chain rule for integrator function with Riemann-Stieltjes integral
I have a random variable $X$ with distribution function $F$. I am interested in evaluating the integral $$ \int g(x) \beta'(F(x)) F(dx) $$ where $\beta$ is smooth and monotone and $g$ is such that the ...
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Finding derivative with two functions sharing the same independent variable
Find $\frac{dy}{dx}$ , provided that $x=f(t)$, $y=g(t)$, $f$ and $g$ are both differentiable, and $y$ is also a differentiable function of $x$, and $f$ is bijective.
already have:
$(h\circ i)'=(h'\...
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Proving that Jacobian of Composition is equal to Composition of Jacobians using epsilon-delta
Let us have functions $\mathrm{f}: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $\mathrm{g}: \mathbb{R}^m \rightarrow \mathbb{R}^k$ such that $\mathrm{f}$ is differentiable at some point $\mathrm{a} \in ...
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Change of coordinates on $\nabla(h\circ\varphi^{-1})$ where $h,\varphi:\mathbb{R}^n\to\mathbb{R}^n$
Say I have a smooth vector-valued function $h:\mathbb{R}^n\to\mathbb{R}^n$ and a smooth diffeomorphism $\varphi:\mathbb{R}^n\to\mathbb{R}^n$. Consider the gradient of the composition $h\circ\varphi^{-...
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