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

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0
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1answer
41 views

Is this sum differentiable w.r.t n? [closed]

Let $T = \sum_{k=1}^n k^m$ Is T differentiable w.r.t $n$ ?
-3
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2answers
55 views

How could a rocket/missle know it's exact speed [With TL;DR]?

So i was reading my book and it seems that it makes a very odd statement, claiming that differentiation is used by rocket scientists to calculate the speed of the rocket. Now that seems very weird to ...
1
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1answer
32 views

Is proof for $\int_0^{x^2} e^{t^2} dt \ge x \sin x$ for $x \ge 0$ correct?

$$F(x)= \int_0^{x^2} e^{t^2} dt$$ show: for each $x \ge 0: \space$ $F(x) \ge x \sin x$ I know the solution that define $G(x)=F(x)-x \sin x \space$ and showing it increasing using derivatives. But ...
0
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1answer
47 views

Deciding where f is differentiable and where it is not.

Am I right to say that the function will not be differentiable at x=1 and x=-1, but differentiable everywhere else? How can I prove those claims?
1
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1answer
35 views

Is f differentiable at 0 for these two functions?

I fail to see how are these two functions different in terms of differentiability. For (a) I would say that f is differentiable at 0 as left and right limits are the same, but what about (b)? How ...
0
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1answer
58 views

Prove that $f(x)=x^n$ is differentiable at every real $x$ and calculate its derivative.

In order to prove that f is differentiable at a point we can consider if the limit (of the basic differentiation expression) exists. However how can I prove that this function is differentiable at ...
0
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1answer
37 views

Is there a mathematical name for the point where $n/x=x$

Were the graph of the asymptotic function $n/x$ rotated counterclockwise about the origin $45^O$, its derivation point would be 0 at $\sqrt{n}$ rotated similarly about the y axis where $n/x=x$. The ...
1
vote
0answers
25 views

Product rule of exponential matrix differentiation

Let $X,Y$ be two $n\times n$ complex matrices. Consider the function $f(t)=e^{tX}e^{tY}$. Is it correct that $\frac{d}{dt}f(t)=Xe^{tX}e^{tY}+e^{tX}Ye^{tY}$? Otherwise, how to prove that $X +Y$ is ...
1
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0answers
23 views

Derivative of vector expression with min

Let vector $f(v) = v*\min(v)$, where $v \in \mathbb{R}^n$. Suppose we want to calculate $\frac{\partial f}{\partial v} = \Big(\frac{\partial f}{\partial v_1}, \dots, \frac{\partial f}{\partial ...
0
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2answers
26 views

Equation of a tangent

I am asked to find the equation of the tangent line (or plane) at the given point. $f(x,y,z)=x^2+2y^2+3z^2=6$ at $(1,-1,1)$. I have computed that $\nabla f(x,y,z)=(2x,4y,6z)=(2,-4,6)$ at ...
1
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0answers
31 views

What do these variables mean in regard to the wave equation and spherical waves?

https://en.wikipedia.org/wiki/Wave_equation#Spherical_waves Before it states ''where K=w/c'', there is an equation that has the following variables: d,r,w,c,l. It also has f_lm(r) What do each of ...
1
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0answers
33 views

approximating function to derivative converge uniformly

I remember reading somewhere that for a $f\in C_c^1(\mathbb{R})$, by definining $$f_n(x) := \frac{f(x+1/n)-f(x)}{1/n},$$ we have that $f_n(x) \longrightarrow f'(x)$ uniformly. I'm pretty sure the ...
1
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1answer
44 views

Derivative of quotient of a formal power series

Suppose $G(x)$ is a formal power series with non-zero constant term. How could I show that $$\Big(\frac{1}{G(x)}\Big)'= -\frac{G'(X)}{G^2(X)}$$
0
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0answers
23 views

Showing that the elasticity of arrival rate for workers wrt $\theta$ is between $0$and $1$

Let $L=E+U$ where $L$ is labour force, $E$ is number of employed, and $U$ is unemployed people. Let $u = \frac{U}{L}$ and $v = \frac{V}{L}$. Given $m(u,v)$ as a matching function that determines ...
1
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2answers
33 views

Looking for proof of a second derivative identity

I'm pretty sure this is true, but haven't been able to figure out or find a proof, largely because I haven't been able to figure out what to Google for. $$ \frac{d^2x}{dy^2} (\frac{dy}{dx})^2+ ...
1
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1answer
40 views

Is there a word describing the derivatives of an object's motion?

Consider an object moving along a straight line. One might say something about its displacement, velocity or acceleration. These are the 0th, 1st and 2nd derivatives of the object's displacement. ...
0
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2answers
25 views

How to solve $\frac {\partial f} {\partial y} = xf$ given $f(x,0) = 1 \ \forall x$?

$f$ is a function of continuous partial derivatives. Find $f$ if: $\dfrac {\partial f} {\partial y} = xf$ $f(x,0) = 1 \ \forall x$
-4
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1answer
42 views

How do I set up this

I am unsure on how to set this up. All I know is I have to use the volume of the cone $\frac{1}{3}\pi r^{2}h.$ Consider a conical tank whose radius at the base is 4 feet, and whose height is 12 feet. ...
4
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0answers
64 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 ...
1
vote
1answer
63 views

Looking for a nonrecursive formula for the general derivatives of the quotient of functions

I want to prove that the $k$-th derivative $h^{(k)}(x)$ of the function $h(x)=\frac{1}{1+x^2}$ is zero at $x=0$ for all integer values $k>0$. My only idea was to go the stubborn way applying ...
0
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1answer
18 views

Solution to parameter dependent heat equation

The Question: Let $p\in [1,\infty)$. I'm looking for a function $f(t)$ such that $u(x,t):=f(t)cos(x)^p$ solves the following heat equation: \begin{align} u_t & = \frac{u_{xx}}{2} \end{align} ...
0
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1answer
14 views

Hairy systems of equation on differentiability of piecewise function

I'm having trouble trying to eliminate and solve for variables that satisfy continuity in a piecewise defined function. The asks for a value of $a$, $b$, and $c$ where the below function is ...
0
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1answer
21 views

Given curve $y=2\tan(\pi x/4)$, find tangent line equation at $1$

Given curve $y=2\tan(\pi x/4)$, find tangent line equation when $x=1$ $$\frac{dy}{dx}= 2\frac \pi 4 \sec^2\left(\frac{\pi x} 4 \right) = \frac{\pi2}{4\cdot2} =\pi$$ when $x=1$ so how do I find the ...
0
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1answer
24 views

Describe the distributional derivative of $f$

Let $f$ be a piece wise defined function with piece wise continuous derivative. Describe the distributional derivative of $f$. My try: If I suppose that the jump discontinuities are at the points ...
1
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1answer
44 views

$f,g$ entire functions , $g$ never vanishes , $f$ is not identically $0$ and $\lim_{|z|\to\infty}f(z)/g(z)$ exists , then $f$ never vanishes?

Let $f,g:\mathbb C \to \mathbb C$ be entire functions such that $g(z)\ne 0,\forall z \in \mathbb C$ , and $\lim_{|z|\to\infty}\dfrac{f(z)}{g(z)}$ exists ; then is it true that either $f(z)=0 , \forall ...
0
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3answers
214 views

Differentiating $4^{x^{x^x}}$

$4^{x^{x^x}}$ Hi, I came across this question and would like to check whether I have it done correctly: $e^{x^3}\ln4=4^{x^3}(3\ln4\cdot x^2)$ is this the correct solution?
1
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0answers
11 views

Reparametrization in regards to Fisher information [duplicate]

I'm trying to understand equation 5.11 from Lehmann and Casella's Theory of Point Estimation (2nd edition) which is presented without proof. It states that if $I(\theta)$ represents the Fisher ...
0
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1answer
37 views

Derivative at Endpoint

In Rudin's "Principles of Mathematical Analysis" he defines the limit of a function as follows. Let $X$ and $Y$ be metric spaces; suppose $E \subset X$, $f$ maps $E$ into $Y$, and $p$ is a limit ...
0
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2answers
88 views

Derivative of $\ln (z), z\in\mathbb{C}$

Let $f(z) = \ln z := \ln |z| + \arg (z)i$. Then the derivative is (if it exists) by definition: $$\lim_{h\to 0}\frac{\ln (z+h)-\ln (z)}{h}=\lim_{h\to 0}\frac{\ln |z+h| +\arg(z+h)i-\ln |z| -\arg(z)i ...
4
votes
2answers
131 views

Show that $f'(0)= \lim_{\Delta x \to 0}\frac{f(\Delta x)-1}{\Delta x} = 1$

This question is related to another question I asked here. Specifically, using the definition of $e$ I gave in that question: There exists a unique complex function $f$ such that $f(z)$ ...
1
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2answers
17 views

given absolute function, find the point where slope=1

$y=|x^2-1|$ where $dy/dx=1$ $dy/dx=2x$ when $y=x^2-1$ $dy/dx=-2x$ when $y=-(x^2-1)$ I managed to find two point after splitting the function into two pieces, but my given answer only accept ...
0
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0answers
35 views

Calculating derivatives for Matlab's NN performance functions

I have a custom Matlab's Neural Network performance function, that is a modification of the MSE function. The modification is that before squaring the errors, it doubles errors that meet certain ...
0
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1answer
21 views

Reparametarize a curve to move a unit length

I'm interested in the general case when we have a curve $(x,f(x))$ parameterized by $x$ to find a parametrization $x=g(t)$ such that $ds/dt=1$ along the curve. So far what I came up: ...
1
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3answers
61 views

Complex derivative of $\cos(x) \cosh(y)-i \sin(x) \sinh(y)$

The problem: Determine the derivative of the following function $f(z)=\cos(x) \cosh(y)-i \sin(x) \sinh(y)$ The original exercise can be found at 2.17 (e) page 36 Should i try to rewrite the function ...
3
votes
2answers
30 views

derivative on endpoints

what's the derivative of $f(x)= x^{2}$ ($x\geq 0$) when x=0? from my understand, it doesn't exist because even $\lim_{h \to 0^{+}}\frac{f(x+h)-f(x)}{h}$ is 0, but $\lim_{h \to ...
1
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1answer
87 views

Proof of existence and uniqueness of the exponential function using ODEs

In our lecture notes for our complex analysis class, we were given the following theorem: Theorem: There exists a unique complex function $f$ such that $f(z)$ is a single valued function ...
0
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2answers
17 views

Evaluate: $\frac{d}{df}\big( \int_{X} f^{2}(x_{1} - x) dx - 2\int_{X} g(x)f(x_{1} - x)dx \big)$

I want to take the derivative of: $\frac{d}{df}\big( \int_{X} f^{2}(x_{1} - x) dx - 2\int_{X} g(x)f(x_{1} - x)dx \big)$ Where $X \subset \mathbb{R}$, and $f,g: \mathbb{R} \rightarrow \mathbb{R}$. ...
2
votes
1answer
26 views

$k-1$st derivative of a degree $k$ polynomial

I know this is going to come across as a very strange question, but it's important that I know the answer. Say I have a degree $k$ polynomial (for my case, I need it to be a complex-valued polynomial ...
0
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1answer
28 views

Proving differentiability at a point

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a continuous function in $\mathbb{R}^n$ and $C^1$ in $\mathbb{R}^n \backslash{0}$ such that $\bigtriangledown f(x)\rightarrow L$ as $x \rightarrow 0$, ...
3
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1answer
18 views

A question on the assumptions in the theorem for changing limit and derivative for sequence of functions

I am looking for an example for a sequence of differentiable functions $\{f_n\}$ on a closed bounded interval $[a,b]$ such that $\{f_n\}$ converges uniformly to a differentiable function $f$ , ...
0
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0answers
15 views

Does a derivative with respect to a function and a derivative with respect to variable commute?

Lets say $\phi:M\to\mathbb{R}$ is a function defined on the manifold $M$ and $\{x^\mu\}$ are the coordinates of some chart $U$ on it. I am trying to check if $ ...
1
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1answer
35 views

Computing the directional derivative of a functional

I'm studying the numerical applications of the total variation using Vogel's "Computational methods forinverse problems", but I'm stuck with some (presumably easy) calculus issues. At a certain point ...
0
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1answer
40 views

Anyone interpretation of the ratio between the derivative and the square of the function?

so I was wondering if anyone has ever heard of any usage, or came across at some point, of interpretation of what does $\frac{f'}{f^2}$ represents for the scalar function? Or equivalently for ...
1
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1answer
48 views

Finite derivative of the harmonic series

In Knuth's Concrete Mathematics he represents the famous quicksort algorithm in computer science as a infinite sum then shows that sum can be simplified to being essentially harmonic. I want to ...
1
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0answers
30 views

determining an integral using only derivative properties [duplicate]

Please note that this question is different from a similiar one as although the basis is the same, the integral that is being calculated is different. let $\alpha′(x)=β(x),β′(x)=α(x) $ and assume ...
0
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1answer
41 views

determining an integral using only derivative properties of two functions

let $\alpha'(x)=\beta(x), \beta'(x)=\alpha(x)$ and assume that $\alpha^2 - \beta^2 = 1$. how would I go about calculating the following anti derivative : $\int (\alpha (x))^5 (\beta(x))^4$d$x$. I ...
2
votes
0answers
42 views

Taylor expansion and Ito when the value function has a non differentiable point

I am trying to solve a multi-period free boundary problem, of an Ornstein–Uhlenbeck process, where each stopping decision at each period adds a different constant (penalty or bonus). Solving with a ...
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2answers
46 views

Find angle that minimizes distance between two angles

Suppose I wish to find an angle $\theta$ such that it minimizes the angular difference between 10$^{\circ}$ and 350$^{\circ}$ regardless of clockwiseness. Naively, I would construct a function $$f = ...
5
votes
3answers
878 views

Is it possible to find inflection points by setting the first derivative to 0?

I have the following $$y = \frac{x^2}{2}-\ln x$$ $$y'= x - \frac1x$$ I learned that inflection points were found by setting the $2^{nd}$ derivative equal to $0$, however, if I do that in this case ...
0
votes
1answer
40 views

Derivative of local flow of vector field at origin

I'm reading one lemma about local flow of vector field, and there's some point in the proof that I don't understand. Here's the lemma and the proof: What I don't understand is the line (line 5 ...