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

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2
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1answer
38 views

If $||\mathbf{J}_f(x) - I_n|| < \frac{1}{2n}$ for all $x \in \mathbb{R^n}$, then $f$ is a diffeomorphism

Suppose that $f: \mathbb{R^n} \rightarrow \mathbb{R^n}$ is a differentiable function such that $||\mathbf{J}_f(x) - I_n|| < \frac{1}{2n}$ for all $x \in \mathbb{R^n}$. (Note that $\mathbf{J}_f$ is ...
1
vote
0answers
14 views

Matrix derivatives for the HJB and ARE relationship

How does one take the derivative of these matrix equations? (Backround:{My professor used them in the proof showing that the Hamilton-Jacobi-equation equivalently solves the free end-point ...
3
votes
1answer
39 views

Directly proving continuous differentiability

Let us say that we want to prove that a function $f: I \to \mathbb{R}$ defined on an open interval $I$ is continuously differentiable on $I$. One way to do this is to establish that $f'(x)$ exists at ...
3
votes
1answer
63 views

Derivatives 1, 2 and 3 and limits

A question for you. Show that if $\lim_{x\to+\infty} x\,f(x)=0$ and $\lim_{x\to+\infty} x\,f''(x)=0$ then $\lim_{x\to+\infty} x\,f'(x)=0$ Thanks ;)
0
votes
2answers
51 views

Can this inequality be solved with Mean value theorem

As my sub-assignment I have to solve inequality: $$ \ln\left(\frac{1}{x} + 1\right) -\frac{1}{x + 1} > 0 $$ If I understood MVT correctly, I should set $g(x)=\ln\left(\frac{1}{x} + 1\right) ...
2
votes
1answer
57 views

Does all function's domain stay the same\expands as we derivate them?

Lets define a funciton $f(x)$ with a domain of, lets say $a>x>b$. If I derivate this function, it's domain will always stay the same or expand? Or it can be "reduced"? Is that mean that $f'(x)$ ...
2
votes
1answer
133 views

How to find 50th derivative of $\left(\dfrac{\sqrt{1-x}}{\sqrt{1+x}}\right)$?

I need to compute 50th derivative of $$\left(\dfrac{\sqrt{1-x}}{\sqrt{1+x}}\right)$$ Of course I would not compute 50 derivatives. I want to find a certain regularity. And what I have: As ...
2
votes
1answer
52 views

How to show that $\lim_{x\to \infty}f'(x)=0$

Let $f$ be a real-valued, bounded, twice differentiable function defined on $(0,\infty)$ with $f'(x)\ge 0$ and $f''(x)\le 0$. Show that $$\lim_{x\to \infty}f'(x)=0$$ I understand $f: (0,\infty) ...
0
votes
1answer
27 views

Missing something about second derivative tests

I'm studying second derivative tests, concavity and inflection points in khan academy ...
1
vote
1answer
155 views

How to take the derivative of a matrix with respect to itself?

Could someone please explain how to take the derivative of matrix with respect to itself? $$\frac{\partial \textbf{X}}{\partial \textbf{X}}$$ where $\textbf{X}$ is an M x N matrix
0
votes
0answers
26 views

Do these statements prove this formula?

$$ \frac{d^n}{dx^n}[g(x)^{f(x)}] = g(x)^{f(x)} B_n(d_1,\cdots,d_n) $$ Calling $$ d_n = \frac{d^n}{dx^n}[ln(g(x))f(x)] $$ Since faa di bruno's formula states $$ \frac{d^n}{dx^n}[f(g(x))] = ...
1
vote
0answers
39 views

Proof Verification for $n2^{n-1} = \sum\limits_{k=1}^n k\binom{n}{k}$

$(1+x)^n = \sum\limits_{k=0}^n\binom{n}{k}x^k$ by binomial theorem $\frac{d}{dx}(1+x)^n =\frac{d}{dx}\sum\limits_{k=0}^n\binom{n}{k}x^k$ $n(1+x)^{n-1} = \sum\limits_{k=1}^n k\binom{n}{k}x^{k-1}$ ...
0
votes
1answer
21 views

Inverse of a function involving a Jacobian.

Why is it true that if the inverse of both $ \tilde{f} $ and $ f $ exists then: $$ \tilde{f}\left(\vec{x}\right) = [Df(x_{0})]^{-1} f(\vec{x}) $$ $$ \implies \tilde{f}^{-1}(\vec{x}) = ...
2
votes
0answers
80 views

Differentiability and Monotonic Functions

I just read proof from Royden of theorem: 'Every Monotonic functions are differentiable almost everywhere.' But proof use Vitali Covering Lemma. But Vitali Covering Lemma is based on fact if we assume ...
6
votes
4answers
121 views

Show that $\lim_{x \to +\infty}\left(f(x)+f'(x)\right)=0 \Rightarrow \lim_{x \to +\infty} f(x)=0$

How to show that $\lim_{x \to +\infty}(f(x)+f'(x))=0 $ implies $\lim_{x \to +\infty} f(x)=0$?
0
votes
1answer
23 views

prove integration formula relating to derivatives

Could any one help me solve this problem ? it is from Apostol's calculus volume 1
2
votes
0answers
40 views

Help differentiate long equation

I need to differentiate the following equation twice with respect to $\alpha$. It is a profile log likelihood equation, where I need the derivatives to get the information matrix. The equation is: ...
3
votes
2answers
70 views

Finite difference method

I wanted to ask something regarding the finite difference approximation. I used the finite difference to calculate the numerical derivatives of my function. The finite difference is given by the ...
3
votes
4answers
96 views

Derivation of the integral

Evaluate $$\large\frac{d}{dx}\int_{0}^{\large\int_0^{e^x}{\cos (s)\,\mathrm ds}}\sec(t^2)\,\mathrm dt$$ I got the answer to be $$e^x\cdot\sec(\sin^2(e^x))\cdot \cos(e^x)$$ but do not know if ...
1
vote
0answers
129 views

Maximum volume of an open box with a square base?

A box with a square base and an open top is to be made. You have 1200cm^2 of material to make it. What is the maximum volume the box could have? Here's what I did: 1200 = x^2+4xz; where x=length of ...
3
votes
1answer
97 views

If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$

Assume that $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$ and $f$ is positive on the interval $(0,1)$ and $0$ at the endpoints. I want to prove that $$\int_0^1 \Big| \frac{f''(x)}{f(x)} ...
0
votes
0answers
85 views

problem with deriving continuity equation

I am studying Aerodynamics, to be more precise, the fundamentals of Aerodynamics. The first law is the continuity equation, for which it is explained in the book that I am using. However, I wished to ...
0
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0answers
22 views

Parametric derivative of $x^2+y^2+sin(4x)+sin(4y)=4$.

I am trying to parametrize $x^2+y^2+sin(4x)+sin(4y)=4$. I need to find a way of taking the intersections between $x^2+y^2+\sin(4x)+\sin(4y)=4$, and $\tan(nx)$, as n increases from $0\le{n}\le{2\pi}$. ...
0
votes
1answer
19 views

$\frac{d}{dx} f(x)$ piecewise defined

my function is defined as follows: $\frac{1}{x} $if $x \not = 0$, $ 1 $ if $ x=0$. Does the $\frac{d}{dx} f(x)$ in $x=0$ exist?
1
vote
4answers
59 views

Trig differentiation

Prove that there is a constant C such that $$ \arcsin{\frac{1-x}{1+x}} + 2\arctan (\sqrt{x}) = C $$ for all $x$ in a certain domain. What is the largest domain on which this identity is true? What ...
1
vote
1answer
101 views

Maple - Substitute into an expression involving derivative

I'm working with some matrices including derivatives like d/dt(x(t)). I need to replace this whole expression {d/dt(x(t))} with something like xdot. I have tried to use "subs", but it seems to refuse ...
0
votes
2answers
39 views

find the derivative of a function with more than one variable

I have a function $g(a)=f_i(x+a(y-x))$ where a$\in$$\Re$ and x,y$\in \Re^d$. How can I find the first and the second derivative of this function? The second part of the exercise is asking me to use ...
0
votes
0answers
85 views

Calculating mixed strategy Nash equilibria: using the derivative?

From roaming around and looking for ways to calculate the mixed strategy Nash equilibrium, I learned that a general way to do it is by determining the probability of choosing a strategy in such a ...
2
votes
0answers
56 views

Show that $f(x)$ is uniform continuity in $(0,1]$

Suppose that $f(x)$ is a continuously differentiable function in $(0,1]$,and $\lim\limits_{x\rightarrow0^{+}}\sqrt{x}f(x)$ exists. Show that $f(x)$ is uniformly continuous on $(0,1]$.
2
votes
0answers
45 views

Looking for differentiable function $f:\mathbb R \to \mathbb R$ whose derivative is nowhere continuous [duplicate]

Does there exist a differentiable function $f: \mathbb R \to \mathbb R$ such that its derivative $f'$ is nowhere continuous ?
0
votes
0answers
20 views

Tips on effectively representing this recurrence relation in a generalized form.

The recurrence relation is $$ y_n = d_1 y_{n-1}+\frac{d}{dx}[y_{n-1}] $$ a good thing to note as well is $$ \frac{d}{dx}[d_n] = d_{n+1} $$ This is terrible to expand out after a good while, The main ...
0
votes
1answer
22 views

The set of differentiability of an extension from half-plane to the plane

This question is related to: Differentiability: Partially Defined Functions Consider a real-valued function $f:\mathbb{H}^2\to\mathbb{R}$. Are there some that admit no extension differentiable in ...
0
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0answers
40 views

Polynomial Root Multiplicity Testing.

I would appreciate some help here. Either a reference or a proof or just a statement that helps me to conduct research of my own. Long ago when I was studying polynomials intently I read about a ...
1
vote
0answers
25 views

$\phi:\mathbb{R}^2\to\mathbb{C}$,$\phi(x,y)=x+iy=z$,$F=\phi^{-1}f\phi$

$\phi:\mathbb{R}^2\to\mathbb{C}$ be a map $\phi(x,y)=x+iy=z$, let $f:\mathbb{C}\to\mathbb{C}$ be the function $f(z)=z^2$ and $F=\phi^{-1}f\phi$ then I need to say which of the following are correct. ...
0
votes
0answers
40 views

Proving a theorem about matrix derivations

Ok, so Im doing some research and I have to understand the following theorem. The theorem states: Let $h$ be a derivation on $Tn(R)$ with $h(e_{ij})=0,\,\, 1\le i \le j \le n$. Then $h=\bar\delta$ ...
0
votes
1answer
48 views

Show that the $n$th derivative of $f(x)$ is zero for all $n \geq 0$. [duplicate]

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)=e^{-\frac{1}{x^2}}$ for $x \neq 0$ and $f(x)=0$ for $x=0$. I want to show that $f^{(n)}(0)=0$ for all $n \ge 0$. That is, the $n$th ...
2
votes
1answer
118 views

$f: \mathbb{R}^2\to \mathbb{R}^2$ is differentiable, and satisfies an inequality that involves its partials - show that f is a bijection.

Suppose that $f: \mathbb{R}^2\to \mathbb{R}^2$ is differentiable, and the partial derivatives of the components $f_1$, $f_2$ satisfy $$max(|\frac{\ df_1}{dx} -1|, |\frac{df_1}{d_y}|, ...
1
vote
0answers
30 views

Nth Derivative of the function [duplicate]

Find the $n^{th}$ derivative of $$f(x) = e^x\cdot x^m$$ If i am not wrong i have following $1^{st}$ Derivative: $e^x\cdot m \cdot x^{m-1} + x^m\cdot e^x$ $2^\text{nd}$ Derivative: $e^x\cdot m \cdot ...
2
votes
0answers
41 views

Find an integrating factor such that $y'=\frac{1-x+y}{x-y}$ is exact

Yet another question of this sort, and hopefully the last. In the previous question I posted, we were lucky enough and the integrating factor was a function of only one variable, the ansatz $\mu_y=0$ ...
4
votes
2answers
37 views

Ordinary differential equations of the form $M(x,y)dx+N(x,y)dy=0$ question

An ODE of the form $M(x,y)dx+N(x,y)dy=0$ is called "good" if $\frac{\partial (M(x,y))}{\partial y}=\frac{\partial (N(x,y))}{\partial x}$ We are given the differential equation ...
1
vote
2answers
67 views

Nth Derivative of a function

Find the $n^{th}$ derivative of $$f(x) = e^x\cdot x^n$$ If i am not wrong i have following $1^{st}$ Derivative: $e^x\cdot n\cdot x^{n-1} + x^n\cdot e^x$ $2^\text{nd}$ Derivative: $e^x\cdot n\cdot ...
0
votes
0answers
23 views

Functional differentiation in Mehta's “Random Matrices”

I'm trying to understand a bit in this book about functional differentiation, which I don't know much about. According to Wikipedia, $\delta F=\int d^n\boldsymbol{r}\frac{\delta F}{\delta ...
4
votes
1answer
67 views

Nth Derivative of a fucntion

Find the $N^{th}$ derivative of $$f(x) = \sqrt{\frac {1-x}{1+x}}$$ I have got $1^{st}$ derivative as: $\frac{-1}{(1-x)^{1/2}(1+x)^{3/2}}$ and $2^{nd}$ derivative as: ...
3
votes
1answer
122 views

If f ' = 0, then f is constant?

I'm a little confused. After finishing the online multi-variable calculus course from the MIT OCW offerings (I wanted to brush up on the subject more concretely, after my Analysis II course), I ...
0
votes
0answers
24 views

directional derivative of convex function always exist

How we can prove that directional derivative of any convex function is exist for any $x\in \text{dom}(f)$ where $\text{dom}(f)=\left\{ x | f(x)<+\infty\right \}$ and directional derivative of $f$ ...
0
votes
0answers
35 views

Holder continuity and gradient

I am trying to prove the implication of differentiability and constancy from Holder continuity. I have: $\frac{\left\lvert f(x)-f(y) \right\rvert}{x-y} \le M|x-y|^{\lambda} \implies \exists g:x ...
3
votes
1answer
47 views

How to calculate $\frac{d}{d y} y'$

Is there a way to evaluate the following expression? $$ \frac{d}{dy} y' $$ where $$ y' = \frac{dy}{dx}.$$
1
vote
2answers
63 views

If $f'(x)$ has a limit as $x\to x_0$, then the function $f$ is differentiable at $x_0$

I've got a question about mathematical analysis of one-variable functions. Assume that we have a function defined for $x \neq x_0$ as composition/sum/product of differentiable functions and also ...
0
votes
1answer
23 views

Possible Points of Local Extrema

I know that we should check critical numbers (points where f'(x) is either zero or not defined) and endpoints (for a closed interval) as possible points of local extrema of f(x). Obviously, all these ...
1
vote
3answers
63 views

Inflection point of $\,f(t) = \frac{1}{1+e^{(-t)}}$

I am trying to calculate the inflection point of the logistic function $f(t) = \dfrac{1}{1+e^{(-t)}}$. According to the definition given in Wikipedia, "A differentiable function has an inflection ...