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

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12 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 ...
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0answers
16 views

Chain rule for the distributional derivative

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
28 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 ...
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4answers
155 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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1answer
15 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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1answer
17 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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0answers
37 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
35 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 ...
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0answers
18 views

Frechet derivatives of norms

I am not sure where to start. I know the definition of Frechet derivatives but how do I use it for norms like these?
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2answers
32 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 ...
12
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1answer
87 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 ...
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0answers
19 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
37 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 ...
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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.
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2answers
44 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 ...
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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: ...
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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0answers
24 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 ...
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1answer
30 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 ...
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0answers
12 views

How do I calculate Jacobian of formula containing quaternions and vectors?

I am facing a problem in robotics where a robot is localized in 3D-space to build up a map simultaneously (see SLAM, e.g. [1]). One approach is to build up a graph of poses $x_i$ and transforms ...
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0answers
23 views

To what Sobolev space does this function belong to?

I am given this function: $$f(x) = e^{- \sqrt{|x|}}$$ and I want to find $k\in \mathbb{N}, \ p \ge 1$ such that $f \in W^{kp} (\mathbb{R})= \{ f \in L^p (\mathbb{R}) \ | \ \forall \alpha \le k: \ ...
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0answers
17 views

Quasi-linear partial differential equations. Solving them.

This is what I have as a quasi-linear partial differential equation:$$u(x_1,...,x_n), \ \ \ \ \sum_{i=1}^{n}A_i(X,u) \frac{\partial u}{\partial x_i}=A_{n+1}(X,u) \ \ \ (1)$$ Then it says let ...
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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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2answers
43 views

Numerical evaluation of first and second derivative

We start with the following function $g: (0,\infty)\rightarrow [0,\infty)$, $$ g(x)=x+2x^{-\frac{1}{2}}-3.$$ From this function we need a 'smooth' square-root. Thus, we check $g(1)=0$, ...
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1answer
41 views

Find the partial derivative of a sphere with equation $x^2+y^2+z^2=4$

We have a sphere with the following equation: $x^2+y^2+z^2=4$ We seek to find the partial derivative, with respect to $x$, of this equation. We think of this equation as a function of three ...
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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3answers
43 views

Limit of derivative does not exist, while limit of difference quotient is infinite

Can anyone show an example of a function $f$ of a real variabile such that $f$ is differentiable on a neighborhood of a point $x_0 \in \mathbb{R}$, except at $x_0$ itself; $f$ is continuous at ...
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0answers
11 views

Derivative with respect to vectors related through a matrix

Consider a function $g: \mathbb{R}^r \to \mathbb{R} $ and two vectors $\mathbf{b} \in \mathbb{R}^r$ and $\mathbf{c} \in \mathbb{R}^m$ such that $\mathbf{c} = \mathbf{A}\mathbf{b}$. If I calculate the ...
2
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2answers
64 views

Can $f''(x)$ exist if $f'(x)$ is undefined?

For example, the piecewise function $ f(x) = \begin{cases} \sqrt{1 - (x + 1)^2} &-2 \leq x \leq 0 \\ -\sqrt{1 - (x - 1)^2} &0 \leq x \leq 2 \end{cases} $ will, at $f(0)$, give $f'(0) = $ ...
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0answers
19 views

Derivative of equation in matrix form

I need to compute first derivatives of the following function $S(w)$ with respect to $w$. Then solve it. The reason behind that is to minimize $S(w)$. $S(w)=\sum_{i=1}^{n} w_i^{1/2} \bigg(y_i - ...
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2answers
18 views

Finding common tangent line to two functions

Sometimes you want to find the common tangent line of two functions. The first thing that comes to mind to a person that is learning basic calculus is that you should equal the derivatives of those ...
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1answer
47 views

Multivariable Calculus, Parametrization and extreme values

I want to find the extreme values of the function $f(x,y,z) = 2x + 2y + z$ under the constraints $x^2+y^2+z^2 \le 2$ and $x^2 + y^2 \le z$ The task is to use a parametrization of the two ...
2
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2answers
64 views

What is the partial derivative of $f(x,y(x))$?

What is the total derivative of $f(x,y(x,z))$ with respect to $x$? Is it $$\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial x}?$$ If this is correct, what is ...
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2answers
66 views

Why is the function continuous at a point which gives the case 0/0?

I have this function : $f(x) = \frac{6x^2+18x+12}{x^2-4}$, the domain is R. How come its graph is continuous at $x = -2$? I know it can be simplified to $\frac{6(x+1)}{x-2}$ ( firstly $f(x) = ...
2
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1answer
43 views

How to compute $[\dot c, X]$ on a manifold?

Consider a smooth curve $c : [0,1] \to M$ and $X \in \mathcal X (M)$. How can one obtain an explicit formula for $[\dot c, X]$? I know the theoretical approach: for every $t \in [0,1]$ there exist a ...
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2answers
98 views

How to differentiate this integral?

Given $$g(x)=\int_{0}^{x} (x-t)e^{t}dt$$ find out $g''(x)$ I thought of using Lebnitz theorem to differentiate it but using Lebnitz I get this $g'(x)=1\cdot (x-x)e^{x}=0$ I don't know how to find ...
0
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2answers
31 views

What is the instantaneous rate of change in the real world?

I can't grasp this concept of an instantaneous change of rate. How could a point on a function graph have a rate of change in the first place? In this moment I just know that it is named the ...
11
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2answers
58 views

Solution to $\frac{d}{d\frac{1}{x}} x$

If I want to solve $$\frac{d}{d\frac{1}{x}} x$$ is my approach correct? As $$\begin{align*} \frac{d}{d\frac{1}{x}}x&=\\ \text{with }\frac{1}{x}&=y\\ ...
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0answers
41 views

How to prove that the following function has a unique mode?

I am trying to prove that the function $$f(\alpha)=n\ln \alpha-n\ln\Big(\sum_{i=1}^{n}t_i^\alpha+\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx\Big)+(\alpha-1)\sum_{i=1}^{n}\ln t_i,$$ where ...
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0answers
49 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 ...
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1answer
31 views

Limit proof of derivative function

$f:[0,+\infty) \rightarrow \mathbb{R}$ twice differentiable. If $f''$is limited and there is $\lim\limits_{x\to \infty}f(x)$, show that $\lim\limits_{x\to \infty}f'(x) = 0$.
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0answers
10 views

Partial derivative of polynomial dependant on previous time values

I have not touched calculus for a few years, and I am not sure what is going on here. Any help would be greatly appreciated :) Essentially, let $p_{t} = \log p(y_{t}|h_{t},h_{t+1})$, where ...
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1answer
25 views

Function whose $n$-th derivative at $x=0$ is $n^3$ / evaluating the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$

I have troubles proving that the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$ represents the function $f(x)=e^x(x^3+3x^2+x)$. My idea was using the identity theorem for power series and the ...
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2answers
27 views

Confusing result obtained taking second derivative of ye^y

I was doing my calculus homework, and one of the questions asked for the first and second derivative of $ye^y=x$, I did the computations and arrived at $-(x+1)^{-2}$, which was a lot neater and ...
4
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2answers
45 views

Prove that $\lim\limits_{x\to\infty} f'(x)=0$

Let $f$ be a function in $(0,\infty)$ such that $f'(x)$ exists. In addition, $\lim\limits_{x\to \infty} f'(x)=L$ (finite) and $f(n)=0$ for every $n \in \Bbb N$. Prove that ...
0
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1answer
17 views

Lagrangian derivative quotient rule?

How do I find the Lagrangian derivative of: $$\frac{D}{Dt} \left(\frac{x}{y^{a}}\right) = 0$$ where a is a constant?
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2answers
32 views

Using definition of derivative to prove that $\lim_{h \to 0} \dfrac{f(x_0+bh)-f(x_0-ch)}{(b+c)h}=f'(x_0) $

I'm studying in preparation for a Mathematical Analysis I examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 4 of 4, part $a$ and graded for ...
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4answers
75 views

Minimum distance between the curves $f(x) =e^x$ and $g(x) =\ln x$ [on hold]

What is the minimum distance between the curves $f(x) =e^x$ and $g(x) = \ln x$? I didn't understand how to solve the problem. Please help me.
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0answers
30 views

Fourier transform calculus of tempered distributions

For example I wanted to ask confirmation of this calculation, if $u \in \mathcal{S}'(\mathbb{R}^n)$ then $\widehat{D^\alpha u} =(2\pi i \xi)^\alpha \widehat{u}$. By definition $\langle \varphi , u ...