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

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13
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351 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
75 views

What's the minimal structure needed to define a notion of derivative?

I know that, for example, to define a limit all you need is the notion of "closeness" generated by a topology; and to define an integral you need a measure function and a sigma-algebra on which it is ...
9
votes
0answers
86 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. ...
7
votes
0answers
116 views

general solution of the equation $\frac{dy}{dx} =\exp(y/x)$

How can i get the general solution of the equation a) $\frac{dy}{dx} = \exp(y/x)$ b) $\frac{dy}{dx} = \exp(x-y)$ and $y=2$ when $x = 0$ I tried b) first: This is a first-order nonlinear ordinary ...
6
votes
0answers
84 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
0answers
91 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
89 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
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0answers
38 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
56 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
233 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
219 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
472 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
374 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
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0answers
26 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
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0answers
54 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
votes
0answers
64 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
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294 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
160 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
38 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
930 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
85 views
+50

Use of Poincare Lemma in solving $\nabla \times \textbf{A}(\textbf{r})=\frac{\textbf{r}}{r^3}$

Let $U = \mathbb{R}^3 \setminus \{(0,0,z) \}$ (ie $\mathbb{R}^3$ with the $z$-axis removed ) and consider $\beta$ on $U$ given by $\displaystyle \beta = \frac{x dy \wedge dz + y dz ...
3
votes
0answers
49 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
58 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
85 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
63 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
46 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
36 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
120 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
votes
0answers
96 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
40 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
45 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
52 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
46 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
270 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
69 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
49 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
83 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 ...
3
votes
0answers
63 views

Trying to find the lyapunov function

I have the system that I want to show the global asymptotic stability of the origin $$\dot{x_1} = x_2 \\ \dot{x_2} = -g(k_1 x_1 + k_2 x _2) $$ where k1 and k2 are positive numbers. Also, $$g(y)y ...
3
votes
0answers
176 views

Partial derivatives using variables after a transformation

I have a transformation $$(x'_1,x'_2)=(f(x_1,x_2), g(x_1,x_2))$$ and I wish to find $$\partial x'_1\over \partial x'_2$$ how might I evaluate this? If it is difficult to find a general expression for ...
3
votes
0answers
244 views

Sign of derivative of a complicated function

EDIT (for bounty): Consider the differential equation $G(p;x,\lambda)p \left[1-\lambda-x(1+\lambda)\right] + x(1+\lambda)p + (1-x)(1-\lambda) \int_{p}^{1} z G'(z;x,\lambda) dz - (1-\lambda) = 0$, ...
3
votes
0answers
271 views

Taking derivative below an integral

I am trying to solve the following question: If $t>0$, then \begin{align*} \int_{0}^{+\infty} e^{-tx} \; dx = \frac{1}{t} \end{align*} Moreover, if $t \geq a > 0$, then $e^{-tx} \leq ...
2
votes
0answers
29 views

Calculate the distance between intersection points of tangents to a parabola

Question Tangent lines $T_1$ and $T_2$ are drawn at two points $P_1$ and $P_2$ on the parabola $y=x^2$ and they intersect at a point $P$. Another tangent line $T$ is drawn at a point between $P_1$ ...
2
votes
0answers
25 views

Finding the flow of a pushforward of vector field (small edit needed as well)

Let $\mathbb{X}$ be the vector field on $\mathbb{R}^2$ given by $$ \mathbb{X}(x,y) = (y,x). $$ Compute the flow $\Phi_t$ of $\mathbb{X}$. Let $F:\mathbb{R}^2\to \mathbb{R}^2$ be the ...
2
votes
0answers
45 views

Directional derivative (Vector)

Given $f:\mathbb{R}^2 \to \mathbb{R}^2$ is a map $f(x,y)=(u(x,y),v(x,y))$ and $\alpha=(\alpha_1,\alpha_2)$ is a point, then how does one show that $f$ is differentiable (or not) in the direction ...
2
votes
0answers
54 views

solving this differential equation for $y$, Is it even possible?

Lets say I have the following: \begin{gather} \frac{(y')^3 + 3 y' y'' + y'''}{(y')^2 + y''} = \sqrt{1+(y')^2}\\ \frac{((y')^3+3y' y'' + y''')^2}{((y')^2 + y'')^2} = 1+(y')^2\\ \frac{(y')^6 + 6 (y')^4 ...
2
votes
0answers
75 views

Project on slope, rates of change, and instantaneous rates of change

I was wondering if someone could look over my work and tell me if I am doing this correctly. Also, I need help on section D. Not understanding what values to substitute into the difference quotient. I ...
2
votes
0answers
26 views

How can we compute the integral of a Laplacian of a radial function over an open ball

Let $B_R\subseteq\mathbb{R}^n$ be an open ball with radius $R>0$ centered at $0$ and $f\in C^0\left(\overline{B_R}\to\mathbb{R}\right)$ be a radial function, i.e. $f(x)=f(r)$ with ...