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

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14
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445 views

Divergence of the Derivative of the Prime Counting Function

On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written $$ \pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1} $$ with $ \operatorname{R}(z) = ...
7
votes
0answers
235 views

Problem with differentiation under integral sign

Original problem: I have a problem in which i need to evaluate the integral: $$ \int_1^\infty \dfrac{\sqrt{r^2-1}e^{-\alpha r}}{r} dr\, $$ I have tried to evaluate it taking the $\alpha$ derivative, ...
7
votes
0answers
235 views

Higher Order Derivative Proof .

I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ ...
7
votes
0answers
110 views

Inverse of a differentiable function equal to its derivative then f is analytic

I've found a nice problem concerning analytic functions. Here it is: Let $f: (0, \infty) \rightarrow \mathbb{R}$ be a function differentiable on $(0, \infty)$ and such that $f^{-1} = f'$. Prove that ...
7
votes
0answers
3k views

Chain rule for matrix - i'm confused

I googled around and searched inside the forum but I'm still confused about a problem. I have 2 matrix functions $f,g : \mathbb{R}^{n \times n} \times \mathbb{R}^{a \times b} \rightarrow ...
6
votes
0answers
83 views

Can it be that $f$ and $g$ are everywhere continuous but nowhere differentiable but that $f \circ g$ is differentiable?

So, I was just asking myself can something like this happen? I was thinking about some everywhere continuous but nowhere differentiable functions $f$ and $g$ and the natural question arose on can the ...
6
votes
0answers
46 views

Generalisation of kth derivative to real values of k

The answer to this question is most likely no, but I'm asking anyway: Assume that $f\in C^n(\mathbb {R,R})$. Is their any natural generalisation of the map $$\{1,2,\ldots,n\}\to C(\mathbb{R, ...
6
votes
0answers
515 views

Differentiation under the Integral Sign

Let $X$ be an open subset of $\mathbb{R}$, and $Y$ be a measure space. Suppose that a function $f:X\times Y\rightarrow \mathbb{R}$ satisfies the following conditions: 1.$f(x,y)$ is a measurable ...
5
votes
0answers
80 views

How should I calculate the $n$th derivative of this expression?

What would be the $n$th derivative of $ f (x) = x ^ x$ I have reached the fifth derivative, very long indeed but I see no pattern that will help me find a general expression. 1 D $y=x^x / ln ...
5
votes
0answers
81 views

partial derivative of a facet normal wrt to one of its vertex

I am struggling to understand the derivation of an equation in a paper (A Bayesian Method for Probable Surface Reconstruction and Decimation, specifically Eqn. 16). Basically they define three ...
5
votes
0answers
63 views

How is $\frac{ds}{dt}$ related to $\frac{dx}{dt}$?

The problem states: Let $x$ and $y$ be differentiable functions of $t$, and let $s = \sqrt{4x^2+6y^2}$ be a function of $x$ and $y$. How is $\frac{ds}{dt}$ related to $\frac{dx}{dt}$ if $y$ is ...
5
votes
0answers
33 views

limit of a region of integration in $\mathbb{R}^2$ approaches a line

I am trying to follow the derivation of derivatives in a paper published in some japanese journal but there seems to be a mistake in the proof. I will present the problem in 2D and in 2 variables so ...
5
votes
0answers
86 views

Equivalence of definitions of $C^k(\overline U)$

let $U$ be an open set of $\mathbb{R}^n$, that contains at least some open set. In Evans book we find the definition $$C^k(\overline U)=\{f \in C^k(U): D^\alpha f \text{ is uniformly continuous on ...
5
votes
0answers
85 views

Leibniz integral rule implementation

Can someone please explain to me why the following expression is true? I really tried to figure out how Leibniz integral rule works, but everytime I think I managed to figure out how to implement it, ...
5
votes
0answers
273 views

Partial derivative of a composite function $\mathbb{R}^n \to \mathbb{R}^n$

I am trying to understand a proof but I am stuck on this technical bit: Apart from the small typo highlighted, I don't really see how to get the big formula for the partial derivative of $v_i$ ...
5
votes
0answers
227 views

Closed form expression for constants

We have the constants $c_{k,n}$ defined by : $$c_{k,n}=\frac{d^{k}}{ds^{k}}\left(\frac{e^{\frac{1}{n(ns-1)}}e^{\psi\left(\frac{s-1}{s} \right )}}{s} \right )$$ Where $\psi(s)\;$ is the Digamma ...
5
votes
0answers
757 views

“Simple” proof of Lebesgue's Differentiation Theorem for dimension 1?

Lebesgue Differentiation Theorem for $\mathbb{R}$: Let $f:[a,b]\to \mathbb{R}$ be intergable and $F(x)=\int_a^xf$. Then $F$ is differentiable almost everywhere in $[a,b]$ and $F'=f$ a.e. Is there a ...
4
votes
0answers
40 views

Find the derivative of the function $ y= x|\cos{\frac{\pi}{x}}|$

Function is defined as it follows : $x \neq 0$ and $f(0)=0$ My work is: $\frac{d}{dx}(x|\cos{\frac{\pi}{x}}|)$ = $|\cos{\frac{\pi}{x}}|$ + $x(\frac{d}{dx}|\cos{\frac{\pi}{x}}|)$ = ...
4
votes
0answers
86 views

Exterior Differential (and its Equivalent Differential Operator) of an Integral 0-Form

I am reading Witten's 1982 paper "Supersymmetry and Morse Theory," and while I am slowly learning the material as I read through the paper, I have come across an equivalence that, while it should be ...
4
votes
0answers
40 views

Linear algebra machinery for differentiation of families of functions.

So I know that since differentiation is linear, for many types of functions we can represent it using linear algebra. Famous examples include polynomials, if we represent them with their coefficients ...
4
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0answers
72 views

Derivatives, discrete and continuous, of $(1/\sqrt{n})\cos (t\log n)$ and $(1/\sqrt{n})\sin (t\log n)$ and Cauchy-Riemann equations

For any arithmetical function $f(n)$, we define its derivative to be $f'(n)=f(n)\cdot \log n$ for $n\geq 1$ (see for example [1], page 45 or Wikipedia). Fact. The functions ...
4
votes
0answers
86 views

Limit and Taylor Series when non-differentiable

I am stuck on a problem similar to this one. Define $$f(\theta,y)=\frac{g(\theta y)}{\int_0^1 g(\theta x)dx}$$ with $g(0)=0$ and $\lim_{t \to 0}g'(t)=+\infty$. I am interested in $\lim_{\theta \to ...
4
votes
0answers
48 views

Prove that the Weierstrass-type function is nowhere differentiable

Given $0<\alpha\leq1$. Show that the function $$f(x)=\sum_{j=1}^\infty 2^{-j\alpha}\sin(2^jx)$$ is nowhere differentiable. I have solved the case $x=0$. Taking $t_l=2^{-l-1}\pi$, then ...
4
votes
0answers
59 views

Find the slope at $t=16$ for $s(t) = $arctan$(\sqrt{t})$

A particle moves along the x axis so that its position at any time when t is greater than or equals zero is $s(t) = $arctan$(\sqrt{t})$. Find the velocity of the particle at $t=16$. The point of ...
4
votes
0answers
251 views

chain rule for derivations

Off we go. So let $b:X\rightarrow Y$ be a function from $X$ to $Y$ endowed with as much structure as it needs to make sense of the question :) and $a:Y\rightarrow \mathbb R$ a function into the reals. ...
4
votes
0answers
60 views

Derivatives of a Dirichlet polynomial

I am new here, so I don't know how this works exactly. If I do something wrong, please let me know. I'd like help to solve a problem I am studying: Let $A$ be finite set of positive integers and ...
4
votes
0answers
137 views

An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
4
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0answers
49 views

Distributions and primitives

I was wondering: if distributions are seen as a generalization of functions that "removes obstructions" to the operation of derivation, is there a generalization of functions that would remove any ...
4
votes
0answers
153 views

The second derivative as a limit

It is well-known that if $f$ is twice differentiable at $a$, then $$ f''(a) = \lim_{h\to 0} \frac{f(a+2h)-2f(a+h) + f(a)}{h^2}. $$ See e.g. this question or this question. On the other hand, the ...
4
votes
0answers
122 views

Taylor Expansions in Spherical Coordinates (Generator of Rotations)

We can expand a smooth function $f:\mathbb{R}^3\to \mathbb{R}$ in a Taylor series: $$f((x^1,x^2,x^3)+(h^1,h^2,h^3))=f(x^1,x^2,x^3)+h_i\frac{\partial f}{\partial x^i}+h_ih_j\frac{\partial^2 f}{\partial ...
4
votes
0answers
479 views

Uniform Differentiability

Let $f:\mathbb{R}^n \to \mathbb{R}$ be differentiable and such that $\nabla f$ is uniformly continuous. Show that $f$ is uniformly differentiable; that is, for any $\epsilon >0$, there is a $\delta ...
4
votes
0answers
196 views

Green's function for third order boundary value problems

How to find the Green's function $G(t,x)$ for the BVP consisting of the equation : $$u'''(t)=0 , \quad t\in (0,1)$$ and BC : $$u(0)=u'(p)=\int_q^1 w(s)u''(s) ds =0 $$ where $\frac12 < ...
4
votes
0answers
43 views

If $D:A\to A$ is a derivation, what can be said about the range of $D$?

What can be said about the relation between the domain and range of a derivation as a function? If $A$ is the domain, any space of functions, what does $D(A)$ look like, where $D$ is a derivation? ...
4
votes
0answers
40 views

Is $H^{\alpha}_{ij,k}=H^{\alpha}_{ik,j}$ for submanifolds in $R^n$?

Consider $\Sigma^n\subset {\bf R}^{n+p}$ a submanifold and $\{e_i, e_\alpha\}$ an orthonormal frame where the Greek indexes stand for the normal vectors. Then define $H^{\alpha}_{ij,k}=e_k\langle ...
3
votes
0answers
42 views

Example of a function differentiable in a point, but not continuous in a neighborhood of the point?

Is there a function that is differentiable in a point $x_0$ (and so continuous of course in $x_0$) but not continuous in a neighborhood of $x_0$ (as said, besides the point $x_0$ itself)? Can anyone ...
3
votes
0answers
28 views

Derivative of product tensor (or matrix product )

if I define a "product tensor" of two second-order tensors as the second-order tensor: $\boldsymbol{C}=\boldsymbol{AB}$ such that $C_{ij}=A_{ik}B_{kj}$ does the product rule applies to ...
3
votes
0answers
52 views

Possible correction to Exercise $5.15$ in Rudin's Principles

Here's Exercise $5.15$ in Rudin's Principles of Mathematical Analysis (Page $115$): Suppose $a \in \mathbb{R}^1$, $f$ is a twice-differentiable real function on $(a,\infty)$, and $M_0$, $M_1$, ...
3
votes
0answers
134 views

Is $h(\mathbf{a},b)=\int_\Omega f(\mathbf{x})e^{-g(\mathbf{x})}\mathrm{d}\mathbf{x}$ differentiable?

Let $f\colon\Bbb{R}^n\to\Bbb{R}$ be an affine function and $g\colon\Bbb{R}^n\to\Bbb{R}$ be a non-negative function. We define $h\colon\Bbb{R}^n\times\Bbb{R}\to\Bbb{R}$ as follows $$ ...
3
votes
0answers
32 views

Is there a general algebraic notion of the chain rule?

To motivate this, I should explain that I have been studying differential fields, i.e. fields endowed with a differentiation operator such that $(a+b)'=a'+b'$ and $(ab)'=a'b+ab'$. Using these rules, ...
3
votes
0answers
22 views

integral form of special function

Do you have any idea to present integral form of this function? $f(x)=\frac{1}{x^2}+\frac{1}{x}(\psi(x)-2\ln x-2)+2(1+\ln x)\psi^{'}(x)+x\ln x\psi^{''}(x).$ Where $\psi^{(n)}(x)$ is polygamma ...
3
votes
0answers
76 views

Study $ h(x)= \sqrt{x^2-1}-x-3$

Let $g$ be the function defined by $$\begin{array}{lrcl} h : & [1;+\infty[ & \longrightarrow & \mathbb{R} \\ & x & \longmapsto & h(x)= \sqrt{x^2-1}-x-3 ...
3
votes
0answers
55 views

Eigenvalues of differentiable matrices

I have a real-valued matrix, $M(a)$, which is a differentiable function of $a$, but not continuously differentiable, with $M(0)=I$. I'll assume $M'(0)$ has distinct eigenvalues. I'm looking for ...
3
votes
0answers
96 views

Differentiation under the integral sign for an electrostatic field

Let $\rho\in C(\bar{D})$ be a continuous function on the compact set $\bar{D}$ and let us define ...
3
votes
0answers
38 views

$f$ is having maxima at $\frac12$ show that $f\circ f$ is having minima at $\frac12$.

$f:[0,1]\to [0,1]$ $f(0)=0=f(1)$ $f$ is having local maxima at $x=\frac12$. Show that $f\circ f(x)$ is having local minima at $x=\frac12$. Using chain rule I was only able to find that $(f\circ ...
3
votes
0answers
35 views

What do the Stirling numbers of the first kind have to do with polylogarithms?

On a whim, I had decided to look into ways of evaluating series of the form $$\sum_{n\ge1}\frac{1}{n^k2^n}$$ which I learned has a more general form in terms of polylogarithms: ...
3
votes
0answers
40 views

Is this a “locally surjective” function?

I quote the "locally surjective" part because I haven't found any reference of that concept, but it kind of fits what I mean. Let $f:\mathbb{R}^N \to \mathbb{R}^M, f \in C^1, x_0 \in \mathbb{R}^N : ...
3
votes
0answers
54 views

gateaux derivative and frechet derivative

In calculus, we have the following equation $DF(x,y)=\partial F_xdx+\partial F_ydy$ if $F$ is differentiable. I think such equation still holds for frechet derivative, but not for gateaux derivative. ...
3
votes
0answers
64 views

Problem about $\lim \limits_{x \to c} f'(x) = l $ implies $f'(c) = l$

I found this problem in a paper. Let $f$ be a function differentiable on $(a, b)$ except possibly at $c \in (a, b)$. Suppose that $\lim \limits_{x \to c} f'(x) = l \in \Bbb R$. Prove that $f$ is ...
3
votes
0answers
30 views

Is there a differentiable function on a closed subset of $\mathbb{R}^n$ that cannot be continued differentiably on an open superset?

Let $A \subseteq \mathbb{R}^n$ be closed with no isolated points and $f:A \to \mathbb{R}^m$. Suppose that for every point $x_0 \in A$ we have (at least one) matrix $L_{x_0}$ such that $$ \lim_{x,y \to ...
3
votes
0answers
70 views

Show that such an $f$ cannot exist

Suppose $f:\mathbb R^n\to\mathbb R$ is a scalar field, such that for a given vector $a\in\mathbb R^n$ and any $y\in\mathbb R^n-\{0\}$ we have, $f'(a;y)>0$. Show that such a function $f$ cannot ...