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

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Frechet Derivatives of normed spaces

(a) Would I use the definition of an open set for one U? How do I show the function is Frechet differentiable. I know the definition but not sure how to apply it. $\lim_{h\to 0}\frac{\lVert ...
3
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1answer
11 views

Which values of $p$, $f$ is it differentiable at the point $(0,0)$?

Let $p \geq 1$ and $f: \mathbb{R^2} \to \mathbb{R}$ defined as $$f(x) = \begin{cases} (\sin \|x\|)^p \cos \frac{1}{\|x\|}, & \quad \text{if } \|x\| \not= 0 \\ 0, & \quad ...
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2answers
45 views

Show that $f$ is not differentiable at $(0,0)$ - $\frac{x_1^2x_2}{x_1^2+x_2^2}$

Let the function $f: \mathbb{R^2} \to \mathbb{R}$ defined as $$f(x) = \begin{cases} \frac{x_1^2x_2}{x_1^2+x_2^2}, & \quad \text{if } (x_1,x_2) \not= 0 \\ 0, & \quad ...
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0answers
9 views

At what points is the norm map Frechet differentiable at

I know the definition of Frechet derivatives but how do I apply it here? Maybe there is a theorem I need to know to make it easier?
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0answers
17 views

Derivative I calculated does not match code (or intuition)?

I want to take the derivative with respect to the $x$ co-ordinate of a Hankel function with the norm of a 2d vector as its argument. Let $\mathbf{x} = (x_1, x_2) \in \mathbb{R}^2$. We have ...
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1answer
19 views

For every normed space the norm map is not Fréchet differentiable at $0$.

Argue that for every normed space $\mathbb{X} \neq \{ 0 \}$ the norm map $\| \ldotp \|_\mathbb{X} : \mathbb{X} \to \mathbb{R}$ is not Fréchet differentiable at $0$. Not really sure where to start ...
35
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2answers
920 views

Function that is the sum of all of its derivatives

I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful ...
2
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1answer
50 views

f: R → R and $|f'(x)| ≤ |f(x)|$ [duplicate]

Let $f: R → R $ be a function such that $f'(x)$ is continuous and $|f'(x)| ≤ |f(x)|$ for all $x ∈ R$ , if $f(0)=0$ the maximum value of $f(5)$ is My Attempt: I proved that $f'(x)=0$ for $x ∈ [0,1]$ ...
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1answer
42 views

Prove that the following function has a unique maximum?

I was working on a problem and reduced it to showing $$f(\alpha)=n\ln \alpha-\ln \left(\sum_{i=1}^n t_i^\alpha+\int_a^b x^{\alpha+\beta-1} e^{-\lambda x^\beta} \, dx \right) + (\alpha-1)\sum_{i=1}^n ...
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1answer
44 views

Limit of a derivative is 1/2 [on hold]

How do I show that $$ \lim_{x \rightarrow b} \frac{d}{dx} \frac{xn^x-bn^b}{n^x-n^b} = \frac{1}{2}$$ where n and b are constants and $n>1$. I saw that it is 1/2 graphing it but I think i still ...
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0answers
64 views

Prove that the following function is convex?

I am trying to prove that the function $$g(\alpha)=\ln\Big(\sum_{i=1}^{n}t_i^\alpha+A(\alpha)\Big) ~~t_i, \alpha>0,$$ where $A(\alpha)=\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx$,is ...
2
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1answer
23 views

Show that $f$ is differentiable at point $x \not= (0,0)$ - $h(x) = (\sin ||x||)^p \cos \frac{1}{||x||}$

Let $p \geq 1$ and $f: \mathbb{R^2} \to \mathbb{R}$ defined as $$f(x) = \begin{cases} (\sin \|x\|)^p \cos \frac{1}{\|x\|}, & \quad \text{if } \|x\| \not= 0 \\ 0, & \quad ...
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1answer
40 views

If given the limit that is a derivative, how do I find it's function and the point? [duplicate]

How would I solve for something like this?? $$\lim_{x\to 5} \frac{2^x - 32}{x-5}$$ using the definition of derivatives.
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3answers
46 views

The given limit is a derivative, but of what function and at what point? [on hold]

How would I solve for something like this?? $$\lim_{h\to 0} \frac{\sqrt[4]{16+h} - 2}{h}$$ using the definition of derivatives.
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1answer
22 views

Problem with convolution, insecure

$$f(t)= t^2\cdot u(t),\quad g(t)=t^4\cdot u(t)$$ I know that I need to use convolution theorem to solve this problem, but I really don't know what to do with step functions. Do I need to include ...
2
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0answers
8 views

Asymptotic distribution for non differentiable functions of estimators

is there kind of a standard tool to derive the distribution of $f(\theta)$ if f is non differentiable (so no Delta Method available) and $\theta$ is asymptotically normal distributed? Thanks a lot!
0
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1answer
15 views

Finding First Integrals in the case $2xy u_x - (x^2+y^2) u_y =0$

Good day, As described in the title, I want to find two First Integrals (FI) to the PDE $$2xy u_x - (x^2+y^2) u_y =0$$ Of course, $u$ is a FI and the solution of the PDE ist $u(x,y)=u_0$. But I want ...
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1answer
880 views

Differentiable function has measurable derivative?

Let $f:[0,T] \to \mathbb{R}$ be a differentiable function. Is it true that $f'$ is measurable? If so, is this also true if $f$ is differentiable almost everywhere? Sorry for lack of effort but I ...
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0answers
11 views

Induced Riemmanian metric and Differential of embedding

Suppose I have a manifold $M$ which is defined as the image of a 1-1 smooth map $G:\mathbb{R}^d\rightarrow H$ into a Hilbert space $H$. I want to understand the Riemmanian metric on $M$ concretely, ...
2
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3answers
5k views

What is the difference between “differentiable” and “continuous”

I have always treated them as the same thing. But recently, some people have told me that the two terms are different. So now I am wondering, What is the difference between "differentiable" and ...
0
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1answer
45 views

Finding Value of C to Maximize Area

f(x)=$xe^{-\sqrt x}$ Find the value of c, such that the area bounded between the graph, the x-axis, x=c, and x=c+1 is maximized. Find the maximum area. I don't know where to start with this one. I ...
4
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4answers
297 views

quick question on an example of the derivative as a linear map.

After reading many answers on the subject I feel like I am close to finally understanding why the derivative is a linear map. I think that if someone helps me understand the following example I might ...
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3answers
27 views

How to find differentiation and integration of curves in general?

Graph of function $f(x)$ How do I go about finding integration and differentiation of curves like these which yield other curves?
2
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1answer
2k views

Minimizing L1 Regularization

I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
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2answers
370 views
+150

Laplacian of a radial function

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a radial function, i.e. $f(x)=f(r)$ with $r:=\left\|x\right\|_2$. As stated at Wikipedia $$\Delta f=\frac{1}{r^{n-1}}\frac{d}{dr}(r^{n-1}f')$$ What's the most ...
0
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0answers
24 views

Chain rule for the distributional derivative [on hold]

Do we have a chain rule for the distributional derivative? My guess is yes, but I do not know how to justify that. Can some one point out how to prove/disprove that? Thanks!
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2answers
39 views

Finding a two-variable function that is distinct from another on every open disk, with specifics.

Consider the two-variable function $$f(x, y) = \sin(x) + \cos(x) + y^2.$$ Find a two-variable function $g(x, y)$ that is distinct from $f(x, y)$ on every open disk which contains the point $(1, 2)$ ...
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2answers
35 views

Can someone explain why $(e,1)$ and $(t, \ln t)$ are the two points of intersection for this question?

I was just going through Khan academy and this question completely threw me. I've rewatched the prior videos a few times to try to understand what I'm suppose to do, but I still don't understand. The ...
4
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3answers
156 views

Failure of differential notation

Through the informal use of differentials, the product rule can be "proved" by writing $$d(fg) = (f + df)(g + dg) - fg = df\,g + f\,dg + df\,dg.$$ Neglecting the product of two differentials, we ...
1
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1answer
18 views

Identity regarding partial derivatives and polar representation

Let $f(x,y)$ be a differentiable function, and $g(r, \theta) = f(r \cos \theta , r \sin \theta)$. I need help showing that: $$ \left( \frac{ \partial f}{\partial x} \right)^2 + \left( \frac{ \partial ...
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1answer
21 views

Show that $\phi_{y,\epsilon} \in D(\mathbb{R^m})$

For a fixed $y$ $$\phi_{y,\epsilon}(x)= \left\{ \begin{array}{ll} \exp(-\frac{\epsilon^2}{\epsilon^2-|x-y|^2}) & \mbox{if $|x-y| \lt \epsilon$};\\ 0 & ...
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0answers
42 views

Compute $\frac{d}{dt}\int_0^t e^{x(s)}ds$, where $x$ is a standard Brownian motion.

How to compute the following differentiation? Is there a general rule that can be applied? $$\frac{d}{dt}\int_0^t e^{x(s)}ds$$ in the case of $x=W$ where $W$ is a standard brownian motion, is there ...
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1answer
39 views

Let $f:(0,1)\to (0,1)$ be a continuously differentiable function. Then which of the following are true?

Let $f:(0,1)\to (0,1)$ be a continuously differentiable function. Then which of the following are true? $1)$ $g=1/f$ is continuous function on $(0,1)$. $2)$ $g=1/f$ is continuously ...
1
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2answers
34 views

Common tangent line to two functions

I have two functions: $$f(x) = x^2 + 3$$ $$g(x) = -x^2 - 2x - 2$$ This two functions have a common tangent line that its slope is positive. My approach: $$f'(x) = 2x$$ $$g'(x) = -2x -2$$ I mark ...
0
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3answers
72 views

Find the derivative of y = $\sqrt{xe^{2x} + 3e^{-x^2}}$

I am trying to find the derivative of this problem but I am not sure where to start. Any help is appreciated. Find the derivative of $$y = \sqrt{xe^{2x} + 3e^{-x^2}}$$
12
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1answer
93 views

Function $\Bbb Q\rightarrow\Bbb Q$ with everywhere irrational derivative

As in topic, my question is as follows: Is there a function $f:\Bbb Q\rightarrow\Bbb Q$ such that $f'(q)$ exists and is irrational for all $q\in\Bbb Q$? For the sake of completeness, I define ...
3
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1answer
247 views
+200

The root of summation function

This is a calculation I need for my statistics project Big edit: simplify the function $f(x)$ a lot. Define for $f(x)$, $x\geq 0$, $$ f(x):=\sum_{k=1}^\infty ...
34
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5answers
3k views

Which derivatives are eventually periodic?

What derivatives are eventually periodic? I have noticed that is $a_{n}=f^{(n)}(x)$, the sequence $a_{n}$ becomes eventually periodic for a multitude of $f(x)$. If $f(x)$ was a polynomial, and ...
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0answers
21 views

Derivative of the maximum of a function on a interval

My question is as follows: Given a function $f: [-h,\infty) \rightarrow \mathcal{R}$ and the maximum function given by \begin{equation} \max_{s\in [-h,0]} |f(t+s)| \end{equation} for $t\geq0$. Then ...
2
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1answer
38 views

Elementary differentiation problem involving logarithms: What am i missing here?

Consider the finite sum $S=$ $\sum_{k=2}^n \log k - \log(k-1)$. Differentiating $S$ w.r.t $k$, we have $S'= \sum_{k=1}^n \dfrac{1}{k} - \dfrac{1}{k-1}=-\sum_{k=1}^n \dfrac{1}{k(k-1)}<0$. But ...
10
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5answers
2k views

Why can a derivative be non-linear?

A definition of the derivative is that it is the slope of the tangent line. For example, $x^3$ has a quadratic derivative. How could the slope of the tangent line be non-linear?
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1answer
37 views

$y=e^x\sin x$; find all points where slope of tangent line equals 0

I have the derivative already. Using the product rule, I got $e^{2}\sin x+e^{2}\cos x$. I can't figure out how to find all the points without graphing it.
8
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1answer
137 views

Find all $f:\mathbb {R} \rightarrow \mathbb {R}$ where $f(f(x))=f'(x)f(x)+c$

Recently, while studying calculus, I have come across multiples problems which asked the following: If $f(x)$ is a polynomial, find all $f(x)$ that $f(f(x))=f'(x)f(x)+c$, where $c$ is a constant. ...
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2answers
50 views

How to find approximate value of $1.01e^{1.01({0.99) }^2} $?

I want to find the approximate value of $1.01e^{1.01({0.99) }^2}$ by using derivative. I tried choosing x=1 and $\delta x=0.01$ it didnt work. How can I start?
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3answers
50 views

Find the derivative of $y=\frac{\tan(x)}{1+\tan(x)}$

$$y=\frac{\tan(x)}{1+\tan(x)}$$ $$\frac{(1+\tan x)(\sec^2x)-(\tan x)(\sec^2x)}{(1+\tan x)^2}$$ I understand this first step but I struggle with simplifying to end up with only $$\sec^2x$$ in the ...
2
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1answer
18 views

General question of derivatives and their inversions

If $\frac{df(x,y)}{dx} = a$, does $\frac{1}{a} = \frac{dx}{df(x,y)}$? Consider $f(x,y) = x^2y \Rightarrow \frac{df(x,y)}{dx} = 2xy \equiv a$, than $\frac{1}{a} = \frac{1}{2xy}$. Now calculate ...
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2answers
77 views

Show a function is Lipschitz

Suppose a real-valued function $f : \mathbb{R} \rightarrow \mathbb{R}$ given by $$f(x) = \begin{cases} \hfill e^{-\frac{1}{\delta^2 - x^2} + \frac{1}{\delta^2}} \hfill & \text{ $|x| ...
0
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0answers
16 views

Integral of dot product of unit vector

I am having trouble with the following integral. $$\int \left(\bar{A} \cdot \hat{ F\left(\lambda\right)}\right)^p\mathrm ds$$ Note that the right hand side of the dot product is normalised. Where: ...
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0answers
25 views

quotient of two differentiable functions is differentiable

I have two functions $k(t)$ and $l(t)$ in a certain closed interval $[a,b]$ both functions are continuous and differentiable in the interval. In addition we have: Both functions are increasing with ...
0
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1answer
46 views

Differentiability of norm

Is a norm from any normed space $\mathbb{A}$ to $\mathbb{R}$ Frechet differentiable at zero? I've tried proving by contradiction that the norm is not Frechet differentiable at zero, but didn't get ...