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Questions tagged [derivatives]

Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

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Prove that $\ell(\Bbb{R}^n,\ell(\Bbb{R}^m,\Bbb{R}^q))=\ell_b(\Bbb{R}^n\times\Bbb{R}^m,\Bbb{R}^q)$

Prove that \begin{align}\ell(\Bbb{R}^n,\ell(\Bbb{R}^m,\Bbb{R}^q))=\ell_b(\Bbb{R}^n\times\Bbb{R}^m,\Bbb{R}^q)\end{align} where $\ell(\Bbb{R}^n,\ell(\Bbb{R}^m,\Bbb{R}^q))$ represents the space of ...
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0answers
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Let $\Phi:ISO(\Bbb{R}^n)\to ISO(\Bbb{R}^n)$, then prove that $\Phi$ is a $C^1-$function and a $C^1-$diffeomorphism

Let $$\Phi:ISO(\Bbb{R}^n)\to ISO(\Bbb{R}^n)$$ I want to prove that $\Phi$ is a $C^1-$function and a $C^1-$diffeomorphism. I know that $\Phi$ is $C^1-$diffeomorphism if $\Phi$ is $C^1$ $\Phi$ is ...
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1answer
7 views

Finding the derivative containing a sum

I'm looking into how linear regression is derived following these steps. I'm curious to how the sum is included in some terms but not others. Just to note, this is a question regarding finding the ...
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1answer
31 views

How to minimize $f(x) = ||Ax-b||$

Solve the problem of minimizing $f(x) = ||Ax-b||$. Consider all the cases and interpret geometrically. If we write $$||Ax-b|| = (a_{11}x_1 + \cdots + a_{1n}x_n - b_1)^2 + \cdots + (a_{n1}x_1 + \...
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2answers
21 views

Existence of a smooth path connecting two points

Let $U=\{(x,y)\in \mathbb{R}^2 | 1<x^2+y^2<4\}$. Let $p,q\in U$. Show that there is a continuous map $\gamma : [0,1] \to U$ such that $\gamma (1)=q$ and $\gamma (0)=p$ and such that $\gamma$ is ...
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31 views

Separation of variables using polar coordinates

I understand that a circle in cartesian coordinates is not a rectangular domain and using polar coordinates gives us our required domain to solve a PDE by separation of variables. However I do not ...
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12 views

Inhomogeneous boundary conditions for partial differential equations

$u_{xx} +u_{yy} = 0$, $(0 < x < a, 0 < y < b)$ $u(0,y)=0$, $u(a,y)=A$ $u(x,0)=0$, $u(x,b)=B$ I need help in understanding how to solve pde's that have inhomogeneous boundary ...
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2answers
39 views

Differentiation of matrix with respect to vector

I have two row vectors $y$ and $k$ of size $1 \times m$ and $1 \times p$ respectively, a matrix $X$ of size $p \times m$. What is the differentiation of: $$(y - kX)^{T} (y - kX)$$ ...
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25 views

Differentiate a function with respect to matrix entries

I am trying to find rotation and translation two sets of points in $\mathbb P^4$ using least squares approach. Say I have a series of points in $P^4$ , $(x_1, x_2,...,x_n)$ and their matching points $(...
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1answer
67 views

Proof explanation: Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$ [duplicate]

Two days ago, I asked a question Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$ but was answered just once. However, I am finding it ...
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1answer
17 views

Continuous function of class $C^2$ that consists of cases $0$ if $x\geq 0$, positive if $x<0$

question I am trying to understand the concept of continuity better and for that I wonder, if the following function $$ f(x) = \begin{cases} 0 &\text{if } x \geq 0\\ x^2 &\text{if } x < 0 ...
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Prove that probability function $Pr(v)=\int_{G(v)} dP$ is smooth

There is a probability density function that depends on non-deterministic ($v$) and random ($x$) parameters: $Pr(v)=\int_{G(v)} dP$, where $G (v)$ is the "goal" region, the probability of getting ...
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1answer
21 views

Proving differentiability of a function

We define the function $f$ in the following manner: $$f(x) = \begin{cases} 0, & \text{if $x = 0$} \\ \vert x \vert^\alpha \text{sin}(\frac{1}{x}), & \text{if $x \neq 0$} \end{cases}$$ Prove ...
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1answer
29 views

Radius of curvature of $r=2 \cdot e^{3 \phi}$

I have to find the curvature of $r=2 \cdot e^{3 \phi}$ $\dfrac{\left[1+\left(\dfrac{d y}{d x}\right)^2\right]^{\frac 3 2}}{\dfrac{d^2 y}{d x^2}}$ $x=r\cdot cos(\phi)=2 \cdot e^{3 \phi}\cdot cos(\phi)...
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1answer
18 views

$\{u_j\}$ an harmonic sequence, $\{\partial^{\alpha}, |\alpha|\le 2\}$ converges uniformly, $u = \lim_j u_j$ is harmonic in $\Omega$

Let $\Omega\subset\mathbb{R}^N$ open and $\{u_j\}$ be a sequence in $C^2(\Omega)$. If each $u_j$ is harmonic in $\Omega$ and the sequences $\{\partial^{\alpha}, |\alpha|\le 2\}$ converge uniformly ...
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0answers
17 views

Functional derivative of gradient of cross product

I need to find functional derivative of the following function with respect to $\eta$ $F = \int[(n\times \nabla\eta) + (m\times \nabla\eta)]^2dr$ Where, n and m are vectors and constant, $\eta$ is a ...
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1answer
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Product rule for 3 or more variables?

I saw another question regarding the product rule for 3(or more) variables here: Finding derivative of three variables I used the second answer in one of my exam papers and got the final answer ...
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0answers
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Plot the graph of the square wave function defined as $f(t)=\sum^{\infty}_{n=0}(-1)^n h_{n}(t)$ on the interval $t>0$ and find its Laplace transform.

Plot the graph of the square wave function defined as $$f(t)=\sum^{\infty}_{n=0}(-1)^n h_{n}(t)$$ on the interval $t>0$ and find its Laplace transform. The http://mathworld.wolfram.com/...
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3answers
45 views

Where do we use continuity?

If $f$ is continuous on $\mathbb{R}$, $f'(0)=1$ and $f(x+y)=f(x)f(y)$ for all $x \in\mathbb{R}$, show that $f'(x)=f(x)$ for all $x\in\mathbb{R}$. Solution: It is clear that $f(0)=1$. For each $x$ we ...
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1answer
33 views

Find the Fourier series of the function: $f(x)=\begin{cases}x+1 & -1\leq x< 0,\\1-x &0\leq x< 1\end{cases},\;\;f(x+2)=f(x)$

I want to find the Fourier series of the function. For now, I am clueless on how to handle the function, int that it has $f(x+2)=f(x).$ $$f(x)=\begin{cases}x+1 & -1\leq x< 0,\\1-x &0\leq x&...
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0answers
30 views

Injective differential of linear operators on a Hilbert space

Given a complex Hilbert space $\mathcal{H}$ of dimension $\dim(\mathcal{H})=d$ and the set $$\mathcal{F} := \{q\in L(\mathcal{H})\mid\text{rank}(q) = 4 \land \lambda^q_{1,2} < 0\ \land \lambda^q_{3,...
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2answers
45 views

Matrix differentiation with respect to matrix

I have an $n \times r$ matrix $A$ and an $r \times m$ matrix $B$. What will the derivative of $ABB^T$ with respect to $A$ be? Is it simply $BB^{T}$?
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1answer
33 views

Prove that $\Vert f(x) -f(y)\Vert\geq (1-k)\Vert x-y\Vert,\;\text{and}\;\Vert f'(x)h\Vert\geq (1-k)\Vert h\Vert,\;\forall\,x,y,h\in\Bbb{R^n}$

Good day all! I'm preparing for a Graduate exam, so I need to solve this problem. Let $f:\Bbb{R}^n\to\Bbb{R}^n$ be a function of class $C^{1}$. We suppose that there exists $k\in ]0,1[$ such that $$\...
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1answer
31 views

Derivatives- Application [on hold]

Assume that spherical raindrop evaporates at a rate proportional to its surface area. If its original radius is 3mm and one hour later, it reduces to 2mm, find an expression for the radius of the ...
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3answers
70 views

Let $f(x^4+8x)=24x$ where $x < 0$, find $f^\prime(0)$

Let $f(x^4+8x)=24x$ where $x < 0$, which of the following equals to $f^\prime(0)$ ? $\text{a)}~1~~~~\text{b)}~-1~~~~\text{c)}~2~~~~\text{d)}~ -2$ This question was asked in high school ...
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1answer
22 views

If $f(x)=e^{-x-y}-x-y-1,\forall\;(x,y)\in\Bbb{R}^2$, then $\exists\;\alpha>0$ and $\varphi:\,]-\alpha,\alpha[\to\Bbb{R}$ of class $\Bbb{C}^{\infty}$

Let $$f:\Bbb{R}^2\to\Bbb{R}$$ $$x\mapsto f(x)=e^{-x-y}-x-y-1,\forall\;(x,y)\in\Bbb{R}^2$$ $i.$ Show that there exists $\alpha>0$ and $\varphi:\,]-\alpha,\alpha[\to\Bbb{R}$ of class $\Bbb{C}^{\...
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1answer
35 views

If $f(x)=\langle x,a\rangle e^{-\langle x, x\rangle} \ \forall x, a \in \Bbb{R}^{n}$, then prove that $f$ is of class $C^{1}$ and compute $f'(x)$

Let $f:\Bbb{R}^n\to\Bbb{R}$ be a function defined by $$f(x) = \langle x, a \rangle e^{-\langle x, x\rangle}$$ $\forall x \in \Bbb{R}^n, \ a \in \Bbb{R}^{n}$. Questions: I want to Prove that $f$ ...
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1answer
40 views

Intuition behind higher derivatives [duplicate]

While I can easily imagine the second derivative conveying the concavity, and the first derivative conveying the slope of any function in a graph. How do I visually understand the meaning of higher ...
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1answer
42 views

Derivative of inverse function proof verification

Can someone verify whether my attempt to prove this theorem is correct? Notice that I use a generalized definition of derivative: Let $f: E \subseteq \mathbb{R} \to \mathbb{R}$ be a function. Let $...
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1answer
67 views

calculating a higher order derivative

My task is to find the values $f^{(2017)}(0)$ and $f^{(2018)}(0)$ for $f(x)=\frac{arccos(x)}{\sqrt{1-x^2}}$. Basically, it's about finding the $n^{th}$ derivative of $f$. So I noticed if I let $g(x)=...
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3answers
72 views

Is there a smarter way to differentiate the function $f(x) = \sin^{-1} \frac{2x}{1+x^2}$?

Given $f(x) = \sin^{-1} \frac{2x}{1+x^2}$, Prove that $$f'(x) = \begin{cases}\phantom{-}\frac{2}{1+x^2},\,|x|<1 \\\\ -\frac{2}{1+x^2},\,|x|>1 \end{cases}$$ Obviously the standard approach ...
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1answer
65 views

What is the series expansion of the $n$-th derivative of this : $\frac{d^n}{dx^n}\int{(e^{-x²})}^{\text{erf}(x)}dx$

$\newcommand{\erf}{\operatorname{erf}}$ The computation of $\frac{d^n}{dx^n}\int{(e^{-x²})}^{\erf(x)}dx$ with wolfram alpha we have for $n=1, n=2, ..n=4$ interesting expansion which seems present ...
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3answers
41 views

How do I take the derivative of $\int g(x - y) \mathrm dx$ with respect to $y$?

How do I take the derivative of $\displaystyle\int g(x - y)\,\mathrm dx$ with respect to $y$? Is it just $-g(x - y)\,\mathrm dx$? Thanks.
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1answer
47 views

Example of convex function which is differentiable, but not twice differentiable?

Are there convex functions for which hessian is not defined, but the gradient is defined everywhere? I was looking at projected gradient descent, as well as Newton's method for solving optimization ...
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1answer
57 views

Does $g (t) $ have a stationary point on $t \in (0,\infty)$? [on hold]

Let $ |\phi(x)| = C\frac {1}{(1 + |x| )^{2}}$ for some $C, $ which implies that $\phi \in L^1$, and let $\int \phi(x) dx =1$. Suppose $f \in L^p (1 \le p \le \infty)$ where $\phi_{t}(x) = t^{-1} \...
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1answer
116 views

Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$

Let $\Omega$ be an open set in $\Bbb{R}^n$ and $f:\Omega\to \Bbb{R}^m $ be of class $\Bbb{C}^2.$ Let $a,b$ be in $\Omega$ such that $[a,b]\subset \Omega.$ Prove that \begin{align}\Vert f(b)-f(a)-f'(a)...
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1answer
42 views

Prove the differentiability of an inverse function, by the theorem of Caratheodory.

The exact statement I am looking forward to prove is that: My try: By the theorem of Caratheodory, there exists a function $\phi:(a,b)\to\mathbb{R}$, $$f(x)-f(m)=\phi(x)(x-m)$$ continuous at $x=m$. ...
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0answers
23 views

Tangent line of a polar curve

I need to find the tangent line where $y=a+x$ (so the tangent line of $45°$) of the following line: $r=\sqrt{\sin(2\cdot\theta)}$ first thing i do i find $x(\theta)$ and $y(\theta)$ $x=\sqrt{\sin(2\...
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2answers
65 views

Determine this limit using L'Hopitals rule

I couldn't find a way to get the answer for $$\lim\limits_{x \to 0} \left(\frac{\sin x}{x}\right)^{\frac{1}{x^{2}}}$$ From my knowledge of L'Hopital's Rule, I see that this is some kind of $1^{\infty}...
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0answers
18 views

Relationship Between CR equations and Complex Differentiability

Consider a complex function $f(x+iy)=u(x,y)+iv(x,y)$ where $u,v$ are real-valued functions. In an attempt to better understand complex differentiation and the Cauchy-Riemann equations, I considered ...
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1answer
36 views

Understanding Taylor Approximations

I am curious about what quantity a Taylor approximation actually optimizes, when it produces, as they say, the "best" possible nth-degree approximation of a function around the given x-value. ...
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0answers
37 views

$f^{(j)}(a) = 0$ except for the last derivative. Under which conditions $a$ can be a minimizer of $f$?

Let $f:\mathbb{R}\to\mathbb{R}$ and suppose that $f^{(j)}(a) = 0, j=0,\cdots,n-1$ and $f^{(n)}(a) \neq 0$. Under which conditions the point $x=a$ can be a minimizer of $f$? Based on your answer: $f(...
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1answer
47 views

On the Definition of the Derivative

How can I show that if a function $f$ from an open interval of the real numbers to an euclidean space is differentiable at a point $x_0$ in its domain, that is, the limit $\lim_{\epsilon \to 0;\ \...
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1answer
29 views

Derivate of vector : transpose, conjugate and conjugate transpose

Let $x$ and $y\in \mathbb{C}^{K\times 1}$ and $H\in \mathbb{C}^{K\times K}$ a diagonal matrix. $\bar{x}$ denotes the complex conjugated, $x^{T}$ denotes the transpose and $x^{*}$ denotes the complex ...
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2answers
41 views

How do I know when it is appropriate to integrate a differential equation?

I am beginning a differential equation textbook, and in the first problem section I encountered a question regarding the change in a population of mice. I am given that $\frac{dP}{dt} = kP^2$, $P(0) = ...
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0answers
22 views

Why complex gradient of $g(z) = \kappa + a^*z + z^*b + z^* C z$ is $\frac{\partial g(z) }{\partial z^*} = b + C z$

I am trying to understand the complex gradient fundamentals here. For example in [1, Example A.3(3)]--also attached the snapshot below for the convenience . How to arrive at the complex gradient of ...
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vote
1answer
37 views

Is the derivative of a nonconstant Lipschitz functions non-zero almost everywhere?

let $U\subset \mathbb{R}^2$ be an open set and $f:U\rightarrow \mathbb{R}$ be Lipschitz continuous, which is not constant. By Rademacher's theorem, the function is differentiable almost everywhere. I ...
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1answer
58 views

If $f: U \to \mathbb{R}^{n}$ is differentiable and $\Vert df(x) \Vert \leq K$ for any $x \in U$ then $\Vert f(x) - f(y) \Vert \leq Kd(x,y)$

Given an open set $U \subset \mathbb{R}^{m}$ connected by paths, consider the metric in $U$ given by $$d(x,y) = \inf_{\gamma \in C(x,y)}\int_{0}^{1}\Vert \gamma'(t) \Vert\text{d}t$$ where $C(x,y)$ ...
1
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3answers
33 views

Strictly decreasing and concave down takes on negative values [duplicate]

Visually it seems that a strictly decreasing twice-differentiable map $f: [0, \infty) \to \mathbb{R}$ with $f''(x) < 0$ for all $x$ should be negative at some point, but I can't come up with a ...
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2answers
47 views

Laplace inverse of $(s+2)U(s)=0$ and $(s+1)U(s)=0.$

I asked this question here and I was given an answer but with some steps unfolded Solve the following problem, $u'(t)+p(t)u(t)=0,\;\;u(0)=0,$ $p(t)=\begin{cases}2& 0\leq t< 1,\\1 &t\geq ...