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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377 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) = ...
9
votes
0answers
92 views

Derivatives of the Struve functions $H_\nu(x)$, $L_\nu(x)$ and other related functions w.r.t. their index $\nu$

There are some known formulae for derivatives of the Bessel functions $J_\nu(x),\,$$Y_\nu(x),\,$$K_\nu(x),\,$$I_\nu(x)\,$with respect to their index $\nu$ for certain values of $\nu$, e.g. ...
6
votes
0answers
118 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, ...
6
votes
0answers
97 views

If the second derivatives $f_{xx}$ and $f_{yy}$ exist, does $f_{xy}$ exist?

If the second derivative with respect to to $x$ exists ($f_{xx}$) and the second derivative with respect to $y$ ($f_{yy}$), does it follow that $f_{xy}$ exists?
6
votes
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121 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 $x_{0}$ ...
6
votes
0answers
91 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 ...
5
votes
0answers
39 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
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0answers
71 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
242 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
222 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
580 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 ...
5
votes
0answers
423 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 ...
4
votes
0answers
35 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
211 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
28 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
54 views

For every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then to prove $f$ is a polynomial in $[a,b]$

Let $f:[a,b] \to \mathbb R$ be a continuous function having derivatives of all order such that for every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then how do I show ...
4
votes
0answers
124 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
32 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
60 views

Intuition on second order partial derivatives

Inspired by smooth submanifolds of $\mathbb{R}^n$, I am looking for a good geometric way to think of second order partial derivatives of a locally smooth function $f:\mathbb{R}^n \rightarrow ...
4
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71 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 ...
4
votes
0answers
352 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
171 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
41 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
1k 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 ...
4
votes
0answers
38 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
27 views

Derivative of a linear basis function over a moving mesh

Given a moving mesh $0=x_0(t)<x_1(t)<\cdots<x_N(t)<x_{N+1}(t)=1,$ where $t$ denotes the current time so that the mesh is moving with time. The linear basis function is then defined as ...
3
votes
0answers
47 views

How can we show that the functions are differentiable?

Show that the following functions $$f(x, y)=\frac{xy}{\sqrt{x^2+y^2}} \\ f(x, y)=\frac{x^2y}{x^4+y^2}$$ are differentiable at each point of the domain. Determine which of them is $C^1$. $$$$ The ...
3
votes
0answers
42 views

Maximum value of an integral.

Define $$f(x)=\int_0^1e^{|t-x|}dt$$ I have to find the maximum value of $f(x)$ when $0 \leq x \leq 1$. To remove the modulus, I wrote $$f(x)=\int_0^xe^{x-t}dt + \int_x^1e^{t-x}dt$$ ...
3
votes
0answers
26 views

Proof of Lipschitz continuous

Wikipedia says that an everywhere differentiable function g : R → R is Lipschitz continuous (with K = sup |g′(x)|) if it has bounded first derivative. How to prove that?
3
votes
0answers
61 views

If $f$ is differentiable at $x_0$, then $\lim_{h\rightarrow 0}(f(x_0 + ah) - f(x_0))/h = af'(x_0)$

If $f$ is differentiable at $x_0$ and $a\in \mathbb R$, show that $ \lim_{h\rightarrow 0} {f(x_0 + ah) - f(x_0) \over h} = af'(x_0)$ EDIT Suppose $u = ah $ then $ h = a/u$. So now we have $$ ...
3
votes
0answers
62 views

Can this summation be expressed differently?

Lets say I have a sum that states the following $$ \sum_{j=0}^{k-c} {k-c \choose j}\ln(a)^{k-c-j} \frac{d^j}{dx^j}[(x)_c] $$ where $(x)_c$ is the falling factorial such that $$ (x)_c = ...
3
votes
0answers
64 views

If $\overline f=f-f'(a)$ then how is $\overline {f'(a)}=0$?

Below is the definition of a function being differentiable at a point, given in my notes: A function $f:A \rightarrow Y$ is said to be differentiable at $a \in A$ if there is a linear map $T \in ...
3
votes
0answers
109 views

Proving $f'(x)<0$ using sequential criterion of limit.

I'm trying to prove the following: Let $f:\mathbb R\rightarrow\mathbb R$ be a function twice differentiable such that $\forall x\in \mathbb R , f(x)>0$ $\forall x\in \mathbb R , f''(x)>0$ ...
3
votes
0answers
64 views

Roots of derivative of q-expontial function

Let the q-deformation of the exponential function be defined by $$ e_q(z)=\sum_{n=0}^\infty{\frac{z^n}{[n]_q!}}. $$ Eq. (1.8) of this paper provides the product representation $$ ...
3
votes
0answers
49 views

Strange things on WolframAlpha: derivation, modulo and doubling result

I asked WA what is the derivative of $\frac1{\cos((x \bmod \pi/2)-\pi/4))}$ equal to for $x=0$. A very strange result came out. The exact result is $-\sqrt2 \mathsf{Mod}^{(1,0)}(0,\frac\pi2)$, ...
3
votes
0answers
88 views

Proof regarding derivatives and Mean Value Theorem.

Original question: $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Show that for any $c \in (a,b)$ that is not a point of maximum or minimum for $f'$, there exist $x_1, x_2 \in (a,b)$ ...
3
votes
0answers
37 views

Prove the following expression is true.

Let $x_1,...,x_{n+1}$ be arbitrary points in $[a,b]$ and let $$Q(x)= \prod\limits_{i=1}^{n+1} (x-x_i)$$Now suppose $f$ is an n times differentiable function and tha P is a polynomial function of ...
3
votes
0answers
133 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 ...
3
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0answers
103 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 ...
3
votes
0answers
42 views

Integration over time by having derivation

Assume we want to find the following integration: \begin{equation}\int_{t=0}^{\infty} p(t)dt\end{equation} where $p(0)=p$ and also $$\frac{dp(t)}{dt}=-p(t)(1-p(t))\mu$$. Is there any easy way to ...
3
votes
0answers
52 views

A vector analysis question requiring multivariable calculus

The question is taken from a Vector-Analysis worksheet as an extension exercise, here is the first part: Let $\underline{v}$ be a vector field $\mathbb{R}^{2} \backslash (0,0)$ of the form ...
3
votes
0answers
45 views

If a positive decreasing function satisfies $f(0)=1$ and $f''\le f$ on $[0,1)$, then $f'(0)\ge-\sqrt{2}$

Let, $f(x)$ be a twice differentiable function defined on $(-1, 1)$ and $f(0) = 1$. Let, $f(x) ≥ 0$, $f'(x) ≤ 0$ and $f''(x) ≤ f(x)$ for all $x ≥ 0$. Show that, $f'(0) ≥ -√2$. I am telling you ...
3
votes
0answers
54 views

Showing that the n first derivatives of (x²-1)^n have at least r roots (for the r-th derivative)?

I have f(x) = (x²-1)^n. I want to show that, for r = 0,1,2,...,n, the r-th derivative is a polynomial (that's easy to show) that has no fewer than r distinct roots in (-1,1). I guess I need to use ...
3
votes
0answers
60 views

Wanted: simple invertible function with specified derivative properties

I'm looking for a positive function $F(x)$, defined for positive real numbers, with the following properties. $F(x)$ is expressible with the standard computer math library routines; $F(x)$ is ...
3
votes
0answers
48 views

Second derivative

I have this functional on $H^1_0$ defined by $J(u)=\frac12||u||^2-\int_0^1 F(t,u(t)) dt $ where $F(t,u(t))=\int_0^u f(t,s) ds $ and i have $J'(u)h= \int_0^1u'(t)h'(t) dt - \int_0^1f(t,u(t)) h(t) ...
3
votes
0answers
275 views

Text with alternative definition of “derivative”?

Instantaneous rates of change are conventionally defined as limits of difference quotients. Rates of things moving at constant speed are definable without delicate issues. If I pass someone moving ...
3
votes
0answers
72 views

Is this answer sufficient to prove? The question is related to second partial derivatives.

Is this answer sufficient to prove ? Does there exist a notation mistake or else? Problem Suppose that the functions $\varphi: \mathbb R \rightarrow \mathbb R$ and $\psi: \mathbb R \rightarrow ...
3
votes
0answers
73 views

Question on derivative

I have this : And i don't understand (3.5) . i.e : why $\displaystyle\frac{d}{dt} G_t(\eta(t)u)=(G'_t(\eta),\eta ')+\partial_tG_t(\eta))$ Please Thank you .
3
votes
0answers
52 views

Derivative of this formula?

I'm studying Solid State Electronics and at one point my book says: $$\dfrac{\text{d}x_n}{\text{d}V_a}= \dfrac{1}{N_d} \left(\dfrac{\varepsilon_s}{2q(\frac{1}{N_a}+\frac{1}{N_d})(\phi_i ...
3
votes
0answers
90 views

The derivative of a family of flows

If one has a family of flows, can one describe the derivative in the "family" direction? Specifically, let $M$ be a smooth manifold and let $X_{s,t}$ be a 2-parameter family of fields on $M$. That ...