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

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11
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328 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
83 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
59 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 $x_{0}$ ...
5
votes
0answers
87 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 ...
5
votes
0answers
36 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
53 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
80 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
221 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
217 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
396 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
23 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
46 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
63 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
96 views

Differentiating $y=x^x$ with the formal definition of a derivative

A friend and I were messing around with derivatives, and while we both know the procedure for finding the derivative of $y=x^x$ with logarithmic differentiation, i.e. $$y=x^x\\ ln(y)=x*ln(x)\\ ...
4
votes
0answers
253 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
149 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
776 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
56 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
59 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
45 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
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0answers
35 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
108 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
82 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
35 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
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0answers
82 views

Related Rates - Differentiation

An airplane is flying at an altitude of 8 miles and passes over a radar station. When the airplane is 12 miles from the base of the station, the radar detects that its horizontal distance is ...
3
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0answers
50 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
43 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
82 views

Is my calculation of $\frac{\partial}{\partial x}\frac{x+y}{\sqrt{y^2-x^2}}$ correct?

Is my calculation of the partial derivative (with respect to $x$) of the function $$f(x,y)=\frac{x+y}{\sqrt{y^2-x^2}}$$ correct? ...
3
votes
0answers
183 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
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0answers
65 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
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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
47 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
82 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
54 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
156 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
233 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
259 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
25 views

Maxima of this function

I am looking for extrema of the function $$g(a,b):=\frac{\pi}{2|b|}\left(\left(\text{C}\left[\frac{a-2 |b|}{\sqrt{2|b| \pi }}\right]-\text{C}\left[\frac{a+2 |b|}{\sqrt{2|b| \pi ...
2
votes
0answers
10 views

Proof that maximal interval of existence exist and bounded

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
2
votes
0answers
55 views

Differentiating $\arcsin(x)$ using first principle method.

Q. Differentiate $\sin^{-1}(x)$ using first principle (delta) method. I did this the following way: $$y=\sin^{-1}(x)$$ $$\therefore\frac{dy}{dx}=\lim_{h\to ...
2
votes
0answers
37 views

How good an approximation to the derivative is an arc-length based approximation?

Note - my original definition below was wrong. I hope this replacement is better. The usual approximation to $f'(x)$ with step size $h$ is $D_h(f, x) = \frac{f(x+h)-f(x)}{h} $. This has so many nice ...
2
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0answers
26 views

Taking partial derivatives over multiple summations

I have the following equation obtained from one of the models. $\mathcal{H} = \sum\limits_{D} \sum\limits_{W}n(d,w)\sum\limits_{Z} p(z|d,w)[\log{p(d)}+\log{p(z|d)}+\log{p(w|z)]}$ I need to take ...
2
votes
0answers
38 views

Determining $f^{-1}(3)$ without knowing $f^{-1}(x)$ but given $f(1)=3$ and $f'(x)>0$.

I have a continuous function $f(x)$ and I want to find $f^{-1}(3)$, but I can't find $f^{-1}$ directly. I know that $f(1)=3$ and $f'(x)>0$ for all x. Because the function is continuous and always ...
2
votes
0answers
32 views

Derivative of the Inverse Cumulative Distribution Function for the Standard Normal Distribution

As the title says, I am trying to find the derivative of the inverse cumulative distribution function for the standard normal distribution. I have this figured out for one particular case, but there ...
2
votes
0answers
76 views

A Tricky Weak Derivatives question

I recently came across the following statement and am having trouble proving it correct. I wonder if you could help. Given a weak derivative, $x'$, there exists an absolutely continuous ...
2
votes
0answers
51 views

$n$th derivative of $f(x)$ using limit definition

After playing around with the limit definition of the derivative for higher order derivatives, I noticed the following odd relationship to determine it for an nth order derivative: Let $F^n=f(x+nh)$ ...
2
votes
0answers
68 views

Definite Integral involving matrices

We have a definite integral of the form given below $ f(t) = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha \tag 1$ Given Data in the question $X(t)$ is a ...