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

learn more… | top users | synonyms (2)

19
votes
0answers
526 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) = ...
8
votes
0answers
116 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
264 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
540 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 ...
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
90 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 ...
6
votes
0answers
47 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, ...
5
votes
0answers
82 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
70 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
270 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. ...
5
votes
0answers
88 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
89 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
277 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
808 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
44 views

Find $f(x)$, when $\left[f''(x)\right]^2\cdot f(x)=\left[f'(x)\right]^3,\ \forall x\in[0,1]$

Let $f:[0,1]\to\mathbb{R}$ be a twice differentiable function with $f''(x)>0,\ \forall x\in[0,1]$, $f(0)=f'(0)=1$ and $\left[f''(x)\right]^2\cdot f(x)=\left[f'(x)\right]^3,\ \forall ...
4
votes
0answers
63 views

Problem on integration: $\int\frac{\log_{\ e}[\sec x]}{\sqrt{1-x^2}}dx$

Well, Today I am very confused over a integral problem and I tried wolfram and many websites they did not help me. Compute: $$\int\frac{\log_{\ e}[\sec x]}{\sqrt{1-x^2}}dx$$ this I have to solve to ...
4
votes
0answers
49 views

Second derivative test for $f(0)=f'(0)=0$ and $f''(0)=2$

Let $f\in C(\mathbb{R}^1)$ with $f(0)=f'(0)=0$ and $f''(0)=2$. Prove that $0$ is strict local minima of $f$. Proof: Since $f''(0)=2$ then $\exists\delta>0$ such that for any $t\in \mathbb{R}^1$ ...
4
votes
0answers
43 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
89 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
45 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
votes
0answers
76 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
99 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
51 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
61 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
62 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
139 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
votes
0answers
53 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
159 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
126 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
264 views

Leibniz's Derivative Rule for Integral in Measure Theory

I saw the extension of Leibniz rule for integrals for measure theory on Wiki, although I am not sure if the proposition there is correct. Besides there is no proof for it. Can anybody please introduce ...
4
votes
0answers
502 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
200 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
43 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
45 views

Why is continuity permissible at endpoints but not differentiability?

Differentiable at endpoints? cause of differentiation only on an open set. Admittedly, there are some questions and answers as to why a function defined on a closed interval is not differentiable on ...
3
votes
0answers
53 views

Simplify an integral formula

I stuck on a statement in the book Mixed Boundary Value Problems - Dean G. Duffy. In page 107, he gives $$ h(t) = \dfrac{2}{\pi} \dfrac{d}{dt}\left\lbrace \int_0^t \dfrac{\cos(x/2)}{\sqrt{\cos(x) - ...
3
votes
0answers
21 views

How to find derivative for $g(x)=\int^{x}_{1}[(f(t))^2-t^2]dt$?

How to find derivative for $g(x)=\int^{x}_{1}[(f(t))^2-t^2]dt$ In my opinion $g'(x)=(f(x))^2-x^2$ and $g'(x)=0$ when $(f(x))^2=x^2$. Is this true?
3
votes
0answers
27 views

holomorphic function on unit disk $D$

Suppose $f$ is holomorphic on $D= (\ z:|z|<1)$ and $f$ is an even function (i.e. $f(z)=f(-z)$). Show that there is a holomorphic function $g$ on $D$ such that $g(x) = f(\sqrt{x})$ for all positive ...
3
votes
0answers
93 views

$41$-th derivative of $\sin(x^2-2x+2)$ at point $1$

Problem: $$\frac{\mathrm d^{41}}{\mathrm dx^{41}}\sin(x^2-2x+2)$$ Can anyone help me with this because I tried to use General Leibniz rule(after I differentiate $\sin(x^2-2x+2)$) and I didn't get ...
3
votes
0answers
31 views

find $\frac{\partial z }{\partial x}$ of tan(xy)+tan(xz)+tan(yz)

Find $\frac{\partial z }{\partial x}$ of $\tan(xy)+\tan(xz)+\tan(yz)=0$ $$\frac{\partial z }{\partial x}\tan(xy)+\frac{\partial z }{\partial x}\tan(xz)+\frac{\partial z }{\partial ...
3
votes
0answers
31 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
62 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
138 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
33 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
24 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
77 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
64 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
99 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 ...