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
20 views

Finding a curve given only its basic form and its tangent line

The basic form is $f(x)=k\sqrt{x}$ and the tangent line is $4x+36$. I've spent over half and hour with wolfram alpha and my notes on this problem: I've tried $(k\sqrt{x})'=4x+36$ and got $k=8\sqrt{x}(...
2
votes
2answers
90 views

How is the second derivitive derived? [closed]

As everyone knows that the derivitive of a function is notated as $\frac{dy}{dx}$ The question is: How is the second derivitive $\left(\frac{d^2y}{dx^2}\right)$ notation derived?
2
votes
1answer
38 views

Show that $fg$ is differentiable at $\hat{x}$ and that $(fg)'(\hat{x})= g(\hat{x})f'(\hat{x}) + f(\hat{x})g'(\hat{x})$

Let $U$ an open set in $\mathbb{R^n}$, $\hat{x} \in U$ and let $f : U \to \mathbb{R}$ and $g : U \to \mathbb{R}$ two different differentiable functions at $\hat{x}$. Show that $fg$ is ...
0
votes
1answer
360 views

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 f(x+h)-f(...
2
votes
1answer
23 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 \...
0
votes
0answers
21 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 $$\frac{\...
0
votes
1answer
576 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 on ...
1
vote
2answers
51 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 \text{...
0
votes
1answer
64 views

Limit of a derivative is 1/2 [closed]

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 don'...
2
votes
1answer
69 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 $x ∈ [0,1]$ ...
1
vote
1answer
79 views

Prove that the following function has a unique maximum? [closed]

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 \...
1
vote
1answer
29 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 \...
-3
votes
1answer
74 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.
0
votes
3answers
95 views

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

How would I solve for something like this?? $$\lim_{h\to 0} \frac{\sqrt[4]{16+h} - 2}{h}$$ using the definition of derivatives.
0
votes
1answer
27 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
votes
0answers
11 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
votes
1answer
24 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 ...
1
vote
0answers
14 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, ...
0
votes
2answers
46 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?
0
votes
1answer
72 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 ...
1
vote
2answers
43 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
votes
4answers
333 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 "...
0
votes
1answer
23 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 & \mbox{otherwise}.\end{...
1
vote
1answer
22 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 ...
1
vote
1answer
65 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 ...
0
votes
1answer
48 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 differentiable ...
1
vote
2answers
38 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
votes
1answer
105 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 $f'...
1
vote
0answers
32 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
votes
1answer
44 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 ...
0
votes
0answers
36 views

minimize this objective function

I have a function to minimize and I don't understand how I should proceed. The function is coming from a publication. Background: In a 2D image, $P_1$ and $P_2$ represents 2 patches of colors (RGB) ...
0
votes
1answer
44 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.
1
vote
2answers
127 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?
0
votes
3answers
56 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 ...
0
votes
0answers
29 views

Integral of dot product of unit vector

I am having trouble with the following integral. $$\int \left(\bar{A} \cdot \hat{ F\left(\lambda\right)}\right)^p\mathrm d\lambda$$ Note that the right hand side of the dot product is normalised. ...
2
votes
1answer
20 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 $\frac{...
1
vote
0answers
48 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
votes
1answer
98 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 ...
1
vote
0answers
22 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 $z_{ij}...
1
vote
0answers
27 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: \ D^{\...
9
votes
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?
2
votes
2answers
51 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 $g(1)=0$, $$g'(x)=1-x^{-\...
1
vote
1answer
82 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 ...
40
votes
6answers
3k views

Which derivatives are eventually periodic?

Which 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 $...
2
votes
3answers
56 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 $x_0$;...
0
votes
0answers
14 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
votes
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) = $ ...
0
votes
0answers
21 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 - \sum_{...
0
votes
2answers
34 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 ...
0
votes
1answer
56 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 constraints/...